1. IntroductionThe Critical 2 d 2 𝑑 2d 2 italic_d Stochastic Heat Flow (SHF) was constructed in [CSZ23a ] as a non-trivial, i.e. non-constant and non-gaussian, solutionto the ill-posed two-dimensional Stochastic Heat Equation (SHE)
∂ t u = 1 2 Δ u + β ξ u , t > 0 , x ∈ ℝ 2 , formulae-sequence subscript 𝑡 𝑢 1 2 Δ 𝑢 𝛽 𝜉 𝑢 formulae-sequence 𝑡 0 𝑥 superscript ℝ 2 \displaystyle\partial_{t}u=\frac{1}{2}\Delta u+\beta\xi u,\qquad t>0,\,x\in%\mathbb{R}^{2}, ∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_u = divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_Δ italic_u + italic_β italic_ξ italic_u , italic_t > 0 , italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (1.1)
where ξ 𝜉 \xi italic_ξ is a space-time white noise. The solution to (1.1 ) lives in the space of generalised functions and, therefore,multiplication is a priori not defined. So in order to construct a solution one has to first regularise the equation. One way to do so isby mollification of the noise ξ ε ( t , x ) := 1 ε 2 ∫ ℝ 2 j ( x − y ε ) ξ ( t , d y ) assign superscript 𝜉 𝜀 𝑡 𝑥 1 superscript 𝜀 2 subscript superscript ℝ 2 𝑗 𝑥 𝑦 𝜀 𝜉 𝑡 d 𝑦 \xi^{\varepsilon}(t,x):=\frac{1}{\varepsilon^{2}}\int_{\mathbb{R}^{2}}j\big{(}%\frac{x-y}{\varepsilon}\big{)}\xi(t,\mathrm{d}y) italic_ξ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_t , italic_x ) := divide start_ARG 1 end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_j ( divide start_ARG italic_x - italic_y end_ARG start_ARG italic_ε end_ARG ) italic_ξ ( italic_t , roman_d italic_y ) , so that (1.1 ) admits a smooth solution u ε superscript 𝑢 𝜀 u^{\varepsilon} italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT , which in fact can also be represented by a Feynman-Kac formula as
u ε ( t , x ) = 𝐄 x [ exp ( β ∫ 0 t ξ ε ( t − s , B s ) d s − β 2 t 2 ‖ j ε ‖ L 2 ( ℝ 2 ) 2 ) ] , superscript 𝑢 𝜀 𝑡 𝑥 subscript 𝐄 𝑥 delimited-[] 𝛽 superscript subscript 0 𝑡 subscript 𝜉 𝜀 𝑡 𝑠 subscript 𝐵 𝑠 differential-d 𝑠 superscript 𝛽 2 𝑡 2 superscript subscript norm subscript 𝑗 𝜀 superscript 𝐿 2 superscript ℝ 2 2 \displaystyle u^{\varepsilon}(t,x)=\boldsymbol{\mathrm{E}}_{x}\Big{[}\exp\Big{%(}\beta\int_{0}^{t}\xi_{\varepsilon}(t-s,B_{s})\mathrm{d}s-\frac{\beta^{2}t}{2%}\|j_{\varepsilon}\|_{L^{2}(\mathbb{R}^{2})}^{2}\Big{)}\Big{]}, italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_t , italic_x ) = bold_E start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT [ roman_exp ( italic_β ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_ξ start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ( italic_t - italic_s , italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) roman_d italic_s - divide start_ARG italic_β start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t end_ARG start_ARG 2 end_ARG ∥ italic_j start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ] , (1.2)
with B s subscript 𝐵 𝑠 B_{s} italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT being a two-dimensional Brownian motion whose expectation when starting from x ∈ ℝ 2 𝑥 superscript ℝ 2 x\in\mathbb{R}^{2} italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is denoted by 𝐄 x subscript 𝐄 𝑥 \boldsymbol{\mathrm{E}}_{x} bold_E start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and j ε ( x ) := 1 ε 2 j ( x ε ) assign subscript 𝑗 𝜀 𝑥 1 superscript 𝜀 2 𝑗 𝑥 𝜀 j_{\varepsilon}(x):=\frac{1}{\varepsilon^{2}}j(\frac{x}{\varepsilon}) italic_j start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ( italic_x ) := divide start_ARG 1 end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_j ( divide start_ARG italic_x end_ARG start_ARG italic_ε end_ARG ) . Then one needs to establish whether a sensible limit can be defined when ε → 0 → 𝜀 0 \varepsilon\to 0 italic_ε → 0 . We will see that for this to be the case a precise choice of β 𝛽 \beta italic_β depending on ε 𝜀 \varepsilon italic_ε will be required, which we will discuss below.
Another approach is by a discretisation scheme; in particular by a distinguished discretisation of theFeynman-Kac formula, which is related to the model ofDirected Polymer in Random Environment (DPRE), [C17 , Z24 ] .The latter is determined by its partition function:
Z M , N β ( x , y ) := E [ exp ( ∑ n = M + 1 N − 1 ( β ω ( n , S n ) − λ ( β ) ) ) 𝟙 { S N = y } | S M = x ] , assign superscript subscript 𝑍 𝑀 𝑁
𝛽 𝑥 𝑦 E delimited-[] conditional subscript superscript 𝑁 1 𝑛 𝑀 1 𝛽 𝜔 𝑛 subscript 𝑆 𝑛 𝜆 𝛽 subscript 1 subscript 𝑆 𝑁 𝑦 subscript 𝑆 𝑀 𝑥 \displaystyle Z_{M,N}^{\beta}(x,y):=\mathrm{E}\Big{[}\exp\big{(}\sum^{N-1}_{n=%M+1}(\beta\omega(n,S_{n})-\lambda(\beta))\Big{)}\mathds{1}_{\{S_{N}=y\}}\,|\,S%_{M}=x\Big{]}, italic_Z start_POSTSUBSCRIPT italic_M , italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( italic_x , italic_y ) := roman_E [ roman_exp ( ∑ start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n = italic_M + 1 end_POSTSUBSCRIPT ( italic_β italic_ω ( italic_n , italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) - italic_λ ( italic_β ) ) ) blackboard_1 start_POSTSUBSCRIPT { italic_S start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = italic_y } end_POSTSUBSCRIPT | italic_S start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT = italic_x ] , (1.3)
where ( S n ) n ≥ 0 subscript subscript 𝑆 𝑛 𝑛 0 (S_{n})_{n\geq 0} ( italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT is a simple, two-dimensional random walk, whose law and expectationare denoted, respectively, by P P \mathrm{P} roman_P and E E \mathrm{E} roman_E and ( ω n , x ) n ∈ ℕ , x ∈ ℤ 2 subscript subscript 𝜔 𝑛 𝑥
formulae-sequence 𝑛 ℕ 𝑥 superscript ℤ 2 (\omega_{n,x})_{n\in\mathbb{N},x\in\mathbb{Z}^{2}} ( italic_ω start_POSTSUBSCRIPT italic_n , italic_x end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N , italic_x ∈ blackboard_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is a family of i.i.d. random variableswith mean 0 0 , variance 1 1 1 1 and finite log-moment generating function λ ( β ) := log 𝔼 [ e β ω ] < ∞ assign 𝜆 𝛽 𝔼 delimited-[] superscript 𝑒 𝛽 𝜔 \lambda(\beta):=\log{\mathbb{E}}\big{[}e^{\beta\omega}\big{]}<\infty italic_λ ( italic_β ) := roman_log blackboard_E [ italic_e start_POSTSUPERSCRIPT italic_β italic_ω end_POSTSUPERSCRIPT ] < ∞ , for β ∈ ℝ 𝛽 ℝ \beta\in\mathbb{R} italic_β ∈ blackboard_R , which serve as the discrete analogue of a space-time white noise.The DPRE regularisation was the one followed in the construction of the Critical 2d SHF in [CSZ23a ] .
In either of the approaches, the singularity that the noise induces in two dimensions demands a particular choiceof the temperature β 𝛽 \beta italic_β , which modulates the strength of the noise. In the DPRE regularisation theCritical 2d SHF emerges through the choice of β = β N 𝛽 subscript 𝛽 𝑁 \beta=\beta_{N} italic_β = italic_β start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT such that
σ N 2 := e λ ( 2 β N ) − 2 λ ( β N ) − 1 = π log N ( 1 + ϑ + o ( 1 ) log N ) , assign superscript subscript 𝜎 𝑁 2 superscript 𝑒 𝜆 2 subscript 𝛽 𝑁 2 𝜆 subscript 𝛽 𝑁 1 𝜋 𝑁 1 italic-ϑ 𝑜 1 𝑁 \displaystyle\sigma_{N}^{2}:=e^{\lambda(2\beta_{N})-2\lambda(\beta_{N})}-1=%\frac{\pi}{\log N}\Big{(}1+\frac{\vartheta+o(1)}{\log N}\Big{)}, italic_σ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := italic_e start_POSTSUPERSCRIPT italic_λ ( 2 italic_β start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) - 2 italic_λ ( italic_β start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT - 1 = divide start_ARG italic_π end_ARG start_ARG roman_log italic_N end_ARG ( 1 + divide start_ARG italic_ϑ + italic_o ( 1 ) end_ARG start_ARG roman_log italic_N end_ARG ) , (1.4)
where o ( 1 ) 𝑜 1 o(1) italic_o ( 1 ) denotes asymptotically negligible corrections as N → ∞ → 𝑁 N\to\infty italic_N → ∞ . In the continuous approximation,β := β ε assign 𝛽 subscript 𝛽 𝜀 \beta:=\beta_{\varepsilon} italic_β := italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ischosen as
β ε 2 = 2 π log 1 ε ( 1 + ϱ + o ( 1 ) log 1 ε ) , subscript superscript 𝛽 2 𝜀 2 𝜋 1 𝜀 1 italic-ϱ 𝑜 1 1 𝜀 \displaystyle\beta^{2}_{\varepsilon}=\frac{2\pi}{\log\tfrac{1}{\varepsilon}}%\Big{(}1+\frac{\varrho+o(1)}{\log\tfrac{1}{\varepsilon}}\Big{)}, italic_β start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT = divide start_ARG 2 italic_π end_ARG start_ARG roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG end_ARG ( 1 + divide start_ARG italic_ϱ + italic_o ( 1 ) end_ARG start_ARG roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG end_ARG ) , (1.5)
where ϱ italic-ϱ \varrho italic_ϱ is given as a function of the above ϑ italic-ϑ \vartheta italic_ϑ and also depends on the mollifier j 𝑗 j italic_j in a particular way. We refer toequation (1.38) in [CSZ19b ] for the precise relation.
The Critical 2 d 2 𝑑 2d 2 italic_d SHF was constructed in [CSZ23a ] as the unique limit of the fields
𝒵 N ; s , t β ( d x , d y ) := N 4 Z [ N s ] , [ N t ] β N ( ⟦ N x ⟧ , ⟦ N y ⟧ ) d x d y , 0 ≤ s < t < ∞ , \displaystyle{\mathcal{Z}}_{N;\,s,t}^{\beta}(\mathrm{d}x,\mathrm{d}y):=\frac{N%}{4}Z_{[Ns],[Nt]}^{\beta_{N}}\left(\llbracket\sqrt{N}x\rrbracket,\llbracket%\sqrt{N}y\rrbracket\right)\mathrm{d}x\mathrm{d}y,\qquad 0\leq s<t<\infty\,, caligraphic_Z start_POSTSUBSCRIPT italic_N ; italic_s , italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( roman_d italic_x , roman_d italic_y ) := divide start_ARG italic_N end_ARG start_ARG 4 end_ARG italic_Z start_POSTSUBSCRIPT [ italic_N italic_s ] , [ italic_N italic_t ] end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( ⟦ square-root start_ARG italic_N end_ARG italic_x ⟧ , ⟦ square-root start_ARG italic_N end_ARG italic_y ⟧ ) roman_d italic_x roman_d italic_y , 0 ≤ italic_s < italic_t < ∞ , (1.6)
where [ ⋅ ] delimited-[] ⋅ [\cdot] [ ⋅ ] maps a real number to its nearest even integer neighbour, ⟦ ⋅ ⟧ delimited-⟦⟧ ⋅ \llbracket\cdot\rrbracket ⟦ ⋅ ⟧ maps ℝ 2 superscript ℝ 2 {\mathbb{R}}^{2} blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT points totheir nearest even integer point on ℤ even 2 := { ( z 1 , z 2 ) ∈ ℤ 2 : z 1 + z 2 ∈ 2 ℤ } assign subscript superscript ℤ 2 even conditional-set subscript 𝑧 1 subscript 𝑧 2 superscript ℤ 2 subscript 𝑧 1 subscript 𝑧 2 2 ℤ {\mathbb{Z}}^{2}_{\text{even}}:=\{(z_{1},z_{2})\in{\mathbb{Z}}^{2}:z_{1}+z_{2}%\in 2{\mathbb{Z}}\} blackboard_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT even end_POSTSUBSCRIPT := { ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∈ blackboard_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT : italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ 2 blackboard_Z } , and d x d y d 𝑥 d 𝑦 \mathrm{d}x\mathrm{d}y roman_d italic_x roman_d italic_y is the Lebesgue measure on ℝ 2 × ℝ 2 superscript ℝ 2 superscript ℝ 2 {\mathbb{R}}^{2}\times{\mathbb{R}}^{2} blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .More precisely,
Theorem 1.1 ([CSZ23a ] ). Let β N subscript 𝛽 𝑁 \beta_{N} italic_β start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT be as in (1.5 ) for some fixed ϑ ∈ ℝ italic-ϑ ℝ \vartheta\in{\mathbb{R}} italic_ϑ ∈ blackboard_R and ( 𝒵 N ; s , t β ( d x , d y ) ) 0 ≤ s < t < ∞ subscript superscript subscript 𝒵 𝑁 𝑠 𝑡
𝛽 d 𝑥 d 𝑦 0 𝑠 𝑡 \big{(}{\mathcal{Z}}_{N;\,s,t}^{\beta}(\mathrm{d}x,\mathrm{d}y)\big{)}_{0\leq s%<t<\infty} ( caligraphic_Z start_POSTSUBSCRIPT italic_N ; italic_s , italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( roman_d italic_x , roman_d italic_y ) ) start_POSTSUBSCRIPT 0 ≤ italic_s < italic_t < ∞ end_POSTSUBSCRIPT be defined as in (1.6 ). Then as N → ∞ → 𝑁 N\rightarrow\infty italic_N → ∞ , the process of random measures( 𝒵 N ; s , t β ( d x , d y ) ) 0 ≤ s ≤ t < ∞ subscript superscript subscript 𝒵 𝑁 𝑠 𝑡
𝛽 d 𝑥 d 𝑦 0 𝑠 𝑡 ({\mathcal{Z}}_{N;s,t}^{\beta}(\mathrm{d}x,\mathrm{d}y))_{0\leq s\leq t<\infty} ( caligraphic_Z start_POSTSUBSCRIPT italic_N ; italic_s , italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( roman_d italic_x , roman_d italic_y ) ) start_POSTSUBSCRIPT 0 ≤ italic_s ≤ italic_t < ∞ end_POSTSUBSCRIPT converges in finite dimensional distributions to a unique limit
𝒵 ϑ = ( 𝒵 s , t ϑ ( d x , d y ) ) 0 ≤ s ≤ t < ∞ , superscript 𝒵 italic-ϑ subscript superscript subscript 𝒵 𝑠 𝑡
italic-ϑ d 𝑥 d 𝑦 0 𝑠 𝑡 \displaystyle\mathscr{Z}^{\vartheta}=(\mathscr{Z}_{s,t}^{\vartheta}(\mathrm{d}%x,\mathrm{d}y))_{0\leq s\leq t<\infty}, script_Z start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT = ( script_Z start_POSTSUBSCRIPT italic_s , italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( roman_d italic_x , roman_d italic_y ) ) start_POSTSUBSCRIPT 0 ≤ italic_s ≤ italic_t < ∞ end_POSTSUBSCRIPT ,
named the Critical 2d Stochastic Heat Flow.
𝒵 ϑ superscript 𝒵 italic-ϑ \mathscr{Z}^{\vartheta} script_Z start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT is a measure valued stochastic process (flow). In fact, its one-time marginals
𝒵 t ϑ ( 𝟙 , d y ) := ∫ x ∈ ℝ 2 𝒵 0 , t ϑ ( d x , d y ) = d 𝒵 t ϑ ( d x , 𝟙 ) := ∫ y ∈ ℝ 2 𝒵 0 , t ϑ ( d x , d y ) , assign superscript subscript 𝒵 𝑡 italic-ϑ 1 d 𝑦 subscript 𝑥 superscript ℝ 2 superscript subscript 𝒵 0 𝑡
italic-ϑ d 𝑥 d 𝑦 superscript 𝑑 superscript subscript 𝒵 𝑡 italic-ϑ d 𝑥 1 assign subscript 𝑦 superscript ℝ 2 superscript subscript 𝒵 0 𝑡
italic-ϑ d 𝑥 d 𝑦 \displaystyle\mathscr{Z}_{t}^{\vartheta}(\mathds{1},\mathrm{d}y):=\int_{x\in%\mathbb{R}^{2}}\mathscr{Z}_{0,t}^{\vartheta}(\mathrm{d}x,\mathrm{d}y)\stackrel%{{\scriptstyle d}}{{=}}\mathscr{Z}_{t}^{\vartheta}(\mathrm{d}x,\mathds{1}):=%\int_{y\in\mathbb{R}^{2}}\mathscr{Z}_{0,t}^{\vartheta}(\mathrm{d}x,\mathrm{d}y), script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( blackboard_1 , roman_d italic_y ) := ∫ start_POSTSUBSCRIPT italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT script_Z start_POSTSUBSCRIPT 0 , italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( roman_d italic_x , roman_d italic_y ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG italic_d end_ARG end_RELOP script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( roman_d italic_x , blackboard_1 ) := ∫ start_POSTSUBSCRIPT italic_y ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT script_Z start_POSTSUBSCRIPT 0 , italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( roman_d italic_x , roman_d italic_y ) , (1.7)
are singular with respect to Lebesgue: it is proven in [CSZ24 ] that if
B ( x , ε ) := { y ∈ ℝ 2 : | y − x | < ε } , assign 𝐵 𝑥 𝜀 conditional-set 𝑦 superscript ℝ 2 𝑦 𝑥 𝜀 B(x,\varepsilon):=\big{\{}y\in\mathbb{R}^{2}\colon\ |y-x|<\varepsilon\big{\}}\,, italic_B ( italic_x , italic_ε ) := { italic_y ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT : | italic_y - italic_x | < italic_ε } , (1.8)
is the Euclidean ball and
𝒵 t ϑ ( B ( x , ε ) ) := ∫ y ∈ B ( x , ε ) 𝒵 t ϑ ( 𝟙 , d y ) , assign superscript subscript 𝒵 𝑡 italic-ϑ 𝐵 𝑥 𝜀 subscript 𝑦 𝐵 𝑥 𝜀 superscript subscript 𝒵 𝑡 italic-ϑ 1 d 𝑦 \displaystyle\mathscr{Z}_{t}^{\vartheta}(B(x,\varepsilon)):=\int_{y\in B(x,%\varepsilon)}\mathscr{Z}_{t}^{\vartheta}(\mathds{1},\mathrm{d}y), script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( italic_B ( italic_x , italic_ε ) ) := ∫ start_POSTSUBSCRIPT italic_y ∈ italic_B ( italic_x , italic_ε ) end_POSTSUBSCRIPT script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( blackboard_1 , roman_d italic_y ) , (1.9)
then for any t > 0 𝑡 0 t>0 italic_t > 0 and ϑ ∈ ℝ italic-ϑ ℝ \vartheta\in\mathbb{R} italic_ϑ ∈ blackboard_R ,
ℙ -a.s. lim ε ↓ 0 𝒵 t ϑ ( B ( x , ε ) ) Vol ( B ( x , ε ) ) = lim ε ↓ 0 1 π ε 2 𝒵 t ϑ ( B ( x , ε ) ) = 0 for Lebesgue a.e. x ∈ ℝ 2 . formulae-sequence ℙ -a.s. subscript ↓ 𝜀 0 superscript subscript 𝒵 𝑡 italic-ϑ 𝐵 𝑥 𝜀 Vol 𝐵 𝑥 𝜀
subscript ↓ 𝜀 0 1 𝜋 superscript 𝜀 2 superscript subscript 𝒵 𝑡 italic-ϑ 𝐵 𝑥 𝜀 0 for Lebesgue a.e. x ∈ ℝ 2 \text{${\mathbb{P}}$-a.s.}\qquad\lim_{\varepsilon\downarrow 0}\;\frac{\mathscr%{Z}_{t}^{\vartheta}\big{(}B(x,\varepsilon)\big{)}}{{\rm Vol}(B(x,\varepsilon))%}=\lim_{\varepsilon\downarrow 0}\frac{1}{\pi\varepsilon^{2}}\mathscr{Z}_{t}^{%\vartheta}\big{(}B(x,\varepsilon)\big{)}=0\quad\text{for Lebesgue a.e.\ $x\in%\mathbb{R}^{2}$}\,. blackboard_P -a.s. roman_lim start_POSTSUBSCRIPT italic_ε ↓ 0 end_POSTSUBSCRIPT divide start_ARG script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( italic_B ( italic_x , italic_ε ) ) end_ARG start_ARG roman_Vol ( italic_B ( italic_x , italic_ε ) ) end_ARG = roman_lim start_POSTSUBSCRIPT italic_ε ↓ 0 end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_π italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( italic_B ( italic_x , italic_ε ) ) = 0 for Lebesgue a.e. italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . (1.10)
The aim of this work is to investigate the intermittency properties of the Critical 2d SHF by studying theinteger moments of the ratio in (1.10 ) and show that, contrary to (1.10 ),they grow to infinity as ε → 0 → 𝜀 0 \varepsilon\to 0 italic_ε → 0 .We also determine the growth rate to be a logarithmic power, up to possible sub-logarithmic corrections.In order to state our result we introduce the notation
𝒵 t ϑ ( φ ) := ∫ ℝ 2 φ ( x ) 𝒵 t ϑ ( d x , 𝟙 ) , assign subscript superscript 𝒵 italic-ϑ 𝑡 𝜑 subscript superscript ℝ 2 𝜑 𝑥 superscript subscript 𝒵 𝑡 italic-ϑ d 𝑥 1 \displaystyle\mathscr{Z}^{\vartheta}_{t}(\varphi):=\int_{\mathbb{R}^{2}}%\varphi(x)\,\mathscr{Z}_{t}^{\vartheta}(\mathrm{d}x,\mathds{1}), script_Z start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_φ ) := ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_φ ( italic_x ) script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( roman_d italic_x , blackboard_1 ) , (1.11)
for any test function ϕ italic-ϕ \phi italic_ϕ on ℝ 2 superscript ℝ 2 \mathbb{R}^{2} blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .Our result then is the following:
Theorem 1.2. Let 𝒰 B ( 0 , ε ) ( ⋅ ) subscript 𝒰 𝐵 0 𝜀 ⋅ {\mathcal{U}}_{B(0,\varepsilon)}(\cdot) caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ( ⋅ ) denote the uniform density on the Euclidean ball in ℝ 2 superscript ℝ 2 \mathbb{R}^{2} blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT :
𝒰 B ( 0 , ε ) ( ⋅ ) := 1 π ε 2 1 B ( 0 , ε ) ( ⋅ ) where B ( 0 , ε ) := { y ∈ ℝ 2 : | y | < ε } , formulae-sequence assign subscript 𝒰 𝐵 0 𝜀 ⋅ 1 𝜋 superscript 𝜀 2 subscript 1 𝐵 0 𝜀 ⋅ assign where 𝐵 0 𝜀 conditional-set 𝑦 superscript ℝ 2 𝑦 𝜀 {\mathcal{U}}_{B(0,\varepsilon)}(\cdot):=\frac{1}{\pi\varepsilon^{2}}\,\mathds%{1}_{B(0,\varepsilon)}(\cdot)\qquad\text{where }\ B(0,\varepsilon):=\big{\{}y%\in\mathbb{R}^{2}\colon\ |y|<\varepsilon\big{\}}\,, caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ( ⋅ ) := divide start_ARG 1 end_ARG start_ARG italic_π italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG blackboard_1 start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ( ⋅ ) where italic_B ( 0 , italic_ε ) := { italic_y ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT : | italic_y | < italic_ε } , (1.12)
and let 𝒵 t ϑ ( 𝒰 B ( 0 , ε ) ) superscript subscript 𝒵 𝑡 italic-ϑ subscript 𝒰 𝐵 0 𝜀 \mathscr{Z}_{t}^{\vartheta}({\mathcal{U}}_{B(0,\varepsilon)}) script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ) be defined as in (1.11 ) with φ ( ⋅ ) = 𝒰 B ( 0 , ε ) ( ⋅ ) 𝜑 ⋅ subscript 𝒰 𝐵 0 𝜀 ⋅ \varphi(\cdot)={\mathcal{U}}_{B(0,\varepsilon)}(\cdot) italic_φ ( ⋅ ) = caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ( ⋅ ) .For all h ≥ 2 ℎ 2 h\geq 2 italic_h ≥ 2 , t > 0 𝑡 0 t>0 italic_t > 0 and ϑ ∈ ℝ italic-ϑ ℝ \vartheta\in\mathbb{R} italic_ϑ ∈ blackboard_R there exist a constant C = C ( h , ϑ , t ) 𝐶 𝐶 ℎ italic-ϑ 𝑡 C=C(h,\vartheta,t) italic_C = italic_C ( italic_h , italic_ϑ , italic_t ) such that
C ( log 1 ε ) ( h 2 ) ≤ 𝔼 [ ( 𝒵 t ϑ ( 𝒰 B ( 0 , ε ) ) ) h ] ≤ ( log 1 ε ) ( h 2 ) + o ( 1 ) , 𝐶 superscript 1 𝜀 binomial ℎ 2 𝔼 delimited-[] superscript superscript subscript 𝒵 𝑡 italic-ϑ subscript 𝒰 𝐵 0 𝜀 ℎ superscript 1 𝜀 binomial ℎ 2 𝑜 1 \displaystyle C\big{(}\log\tfrac{1}{\varepsilon}\big{)}^{h\choose 2}\leq{%\mathbb{E}}\Big{[}\Big{(}\mathscr{Z}_{t}^{\vartheta}({\mathcal{U}}_{B(0,%\varepsilon)})\Big{)}^{h}\Big{]}\leq\big{(}\log\tfrac{1}{\varepsilon}\big{)}^{%{h\choose 2}+o(1)}, italic_C ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) start_POSTSUPERSCRIPT ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) end_POSTSUPERSCRIPT ≤ blackboard_E [ ( script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ] ≤ ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) start_POSTSUPERSCRIPT ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) + italic_o ( 1 ) end_POSTSUPERSCRIPT , (1.13)
with o ( 1 ) 𝑜 1 o(1) italic_o ( 1 ) representing terms that go to 0 0 as ε → 0 → 𝜀 0 \varepsilon\to 0 italic_ε → 0 .
We note that for h = 2 ℎ 2 h=2 italic_h = 2 the correlation structure of the Critical 2d SHF already providesthe sharp asymptotic
𝔼 [ ( 𝒵 t ϑ ( g ε 2 ) ) 2 ] ∼ C t log 1 ε , as ε → 0 , similar-to 𝔼 delimited-[] superscript superscript subscript 𝒵 𝑡 italic-ϑ subscript 𝑔 superscript 𝜀 2 2 subscript 𝐶 𝑡 1 𝜀 as ε → 0
\displaystyle{\mathbb{E}}\Big{[}\Big{(}\mathscr{Z}_{t}^{\vartheta}(g_{%\varepsilon^{2}})\Big{)}^{2}\Big{]}\sim C_{t}\log\tfrac{1}{\varepsilon},\qquad%\text{as $\varepsilon\to 0$}, blackboard_E [ ( script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( italic_g start_POSTSUBSCRIPT italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ∼ italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG , as italic_ε → 0 , (1.14)
see relation (1.21) in [CSZ19b ] .
Moments of the Critical 2d SHF field can be expressed in terms of the Laplace transform of the total collision timeof a system of independent Brownian motions with a critical delta interaction. This is associated to the Hamiltonian− Δ + ∑ 1 ≤ i < j ≤ h δ 0 ( x i − x j ) Δ subscript 1 𝑖 𝑗 ℎ subscript 𝛿 0 subscript 𝑥 𝑖 subscript 𝑥 𝑗 -\Delta+\sum_{1\leq i<j\leq h}\delta_{0}(x_{i}-x_{j}) - roman_Δ + ∑ start_POSTSUBSCRIPT 1 ≤ italic_i < italic_j ≤ italic_h end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) on ( ℝ 2 ) h superscript superscript ℝ 2 ℎ (\mathbb{R}^{2})^{h} ( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT known as the delta-Bose gas [AFH+92 , DFT94 , DR04 ] ; δ 0 ( ⋅ ) subscript 𝛿 0 ⋅ \delta_{0}(\cdot) italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( ⋅ ) is the delta-funtion.This operator is singular and ill-defined due to the delta function. To regularize it, one approach similarto that used for the SHE can be applied, involving a limiting sequence of operators− Δ + ∑ 1 ≤ i < j ≤ h β ε 2 δ ε ( x i − x j ) Δ subscript 1 𝑖 𝑗 ℎ superscript subscript 𝛽 𝜀 2 subscript 𝛿 𝜀 subscript 𝑥 𝑖 subscript 𝑥 𝑗 -\Delta+\sum_{1\leq i<j\leq h}\beta_{\varepsilon}^{2}\delta_{\varepsilon}(x_{i%}-x_{j}) - roman_Δ + ∑ start_POSTSUBSCRIPT 1 ≤ italic_i < italic_j ≤ italic_h end_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) on ( ℝ 2 ) h superscript superscript ℝ 2 ℎ (\mathbb{R}^{2})^{h} ( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ,where β ε 2 superscript subscript 𝛽 𝜀 2 \beta_{\varepsilon}^{2} italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is as in (1.5 ) and δ ε subscript 𝛿 𝜀 \delta_{\varepsilon} italic_δ start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT a mollification of the delta function with a j ε subscript 𝑗 𝜀 j_{\varepsilon} italic_j start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT as in (1.2 ). [DFT94 ] employs, instead, a regularisation in Fourier space.The term critical delta interaction refers to the constant in β ε 2 superscript subscript 𝛽 𝜀 2 \beta_{\varepsilon}^{2} italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT in (1.5 ) being equal to 2 π 2 𝜋 2\pi 2 italic_π .It is well known that independent Brownian motions in dimension 2 2 2 2 do not meet, however, when their joint measure istilted through a critical delta-attraction between them, then in the limit when the regularisation is removed, they do meet and have a nontrivial collision time.This has been demonstrated in [CM24 ] , where it has been established that the local collision time in the case oftwo independent Brownian motions (corresponding to h = 2 ℎ 2 h=2 italic_h = 2 in our setting) has a positive log – Hausdorff dimension.
Our approach to obtaining the bounds in Theorem 1.13 involves expanding the Laplacetransform of the total collision time of h ℎ h italic_h independent Brownian motionsin terms of diagrams of pairwise interactions (see Figure 1 ). Estimating adiagram of this form was first done in [CSZ19b ] * * * more precisely, in [CSZ19b ] the discrete case of independent two-dimensional random walks was treated but the scaling limit recovers the Brownian situation in the case when the starting points of the Brownian motion are spread out rather than beingconcentrated in a ε 𝜀 \varepsilon italic_ε -ball as we study here. Higher-order collision diagrams were estimated in [GQT21 ] , againin the situation of spread out initial points, using an alternative approach, which wasbased on resolvent methods and inspired by [DFT94 , DR04 ] . For sub-critical delta interactions,higher-order collision diagrams of simple two-dimensional random walks were treated in [CZ23 , LZ23 , LZ24 ] .In particular, in [CZ23 ] , collision diagrams involving a number of walks growing up to a rate proportional to the square root of the logarithm of the time horizon were analyzed. In all these cases† † † [LZ23 ] addresses a slightly different setting , collision diagrams express moments of either the stochastic heatequation or the directed polymer model and all of them address scenarios where moments remainbounded. In contrast, here we study the situation where moments blow up in the limit as the size of the ballsε → 0 → 𝜀 0 \varepsilon\to 0 italic_ε → 0 .The lower bound in Theorem 1.2 is reduced to the Gaussian correlation inequality [R14 , LM17 ] –a tool already used in the context of the SHE in [F16 , CSZ23b ] . The upper bound is more demanding as one needsto control the complicated recursions emerging from the collision diagrams. Towards this we were guided by the approach of [CZ23 ] , which was developed to treat the subcritical case. A number of twists have been necessary in order to deal with thesingularities of the critical case, which include introducing suitable Laplace multipliers, optimisationand specific combinatorics.Our theorem leaves open whether higher moments grow as ε → 0 → 𝜀 0 \varepsilon\to 0 italic_ε → 0 proportionally to( log 1 ε ) ( h 2 ) superscript 1 𝜀 binomial ℎ 2 (\log\tfrac{1}{\varepsilon})^{h\choose 2} ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) start_POSTSUPERSCRIPT ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) end_POSTSUPERSCRIPT , i.e. up to a constant factor,or whether there are sub-logarithmic corrections that lead to 1 ( log 1 ε ) ( h 2 ) 𝔼 [ ( 𝒵 t ϑ ( 𝒰 B ( 0 , ε ) ) ) h ] → ∞ → 1 superscript 1 𝜀 binomial ℎ 2 𝔼 delimited-[] superscript superscript subscript 𝒵 𝑡 italic-ϑ subscript 𝒰 𝐵 0 𝜀 ℎ \tfrac{1}{(\log\tfrac{1}{\varepsilon})^{h\choose 2}}{\mathbb{E}}\Big{[}\Big{(}%\mathscr{Z}_{t}^{\vartheta}({\mathcal{U}}_{B(0,\varepsilon)})\Big{)}^{h}\Big{]%}\to\infty divide start_ARG 1 end_ARG start_ARG ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) start_POSTSUPERSCRIPT ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) end_POSTSUPERSCRIPT end_ARG blackboard_E [ ( script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ] → ∞ ; our upper bound includes corrections of order| log ε | 1 | log log log ε | superscript 𝜀 1 𝜀 |\log\varepsilon|^{\frac{1}{|\log\log\log\varepsilon|}} | roman_log italic_ε | start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG | roman_log roman_log roman_log italic_ε | end_ARG end_POSTSUPERSCRIPT .If the former fact is valid, then,together with (1.14 ), this would suggest that pairwise collisions are nearly independent (but will still exhibit a positive correlation) even under a critical delta attraction.Independence of the collision times in the case of subcritical strength of attraction(in a random walk setting) was established in [LZ24 ] .On the other hand, sub-logarithmic corrections would suggest a more intricate correlation structure.In such a case, more refined methods for establishinglower bounds would be necessary. In the subcritical case, lower bounds(within the directed polymer framework), not relying on the Gaussian Correlation Inequality but also slightly less sharp,were derived in [CZ24 ] .Further investigation into this topic would be interesting, and we hope to explore this in the future.Before closing this introduction let us make a connection between our results and the notion of multifractality ,see [BP24 ] . The fractal spectrum of random measure μ 𝜇 \mu italic_μ on ℝ d superscript ℝ 𝑑 \mathbb{R}^{d} blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is determined via the exponent ξ ( h ) 𝜉 ℎ \xi(h) italic_ξ ( italic_h ) inthe moment asymptotics
𝔼 [ μ ( B ( x , ε ) ) h ] ∼ ε ξ ( h ) , as ε → 0 , similar-to 𝔼 delimited-[] 𝜇 superscript 𝐵 𝑥 𝜀 ℎ superscript 𝜀 𝜉 ℎ as ε → 0
\displaystyle{\mathbb{E}}\big{[}\mu(B(x,\varepsilon))^{h}\big{]}\sim%\varepsilon^{\xi(h)},\qquad\text{as $\varepsilon\to 0$}, blackboard_E [ italic_μ ( italic_B ( italic_x , italic_ε ) ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ] ∼ italic_ε start_POSTSUPERSCRIPT italic_ξ ( italic_h ) end_POSTSUPERSCRIPT , as italic_ε → 0 ,
for h ∈ [ 0 , 1 ] ℎ 0 1 h\in[0,1] italic_h ∈ [ 0 , 1 ] . This notion is useful in determining the Hausdorff dimension of the support of the measure andhighlight localisation (see [BP24 ] for further information).The measure μ 𝜇 \mu italic_μ is said to exhibit multifractality if the exponent ξ ( h ) 𝜉 ℎ \xi(h) italic_ξ ( italic_h ) is a nonlinearfunction of h ℎ h italic_h . Our result that
𝔼 [ 𝒵 t ϑ ( B ( x , ε ) ) h ] ∼ ε 2 h ( log 1 ε ) h ( h − 1 ) 2 + o ( 1 ) , as ε → 0 for 2 ≤ h ∈ ℕ , similar-to 𝔼 delimited-[] superscript subscript 𝒵 𝑡 italic-ϑ superscript 𝐵 𝑥 𝜀 ℎ superscript 𝜀 2 ℎ superscript 1 𝜀 ℎ ℎ 1 2 𝑜 1 as ε → 0 for 2 ≤ h ∈ ℕ
\displaystyle{\mathbb{E}}\big{[}\mathscr{Z}_{t}^{\vartheta}\big{(}B(x,%\varepsilon)\big{)}^{h}\,\big{]}\sim\varepsilon^{2h}\,\big{(}\log\tfrac{1}{%\varepsilon}\big{)}^{\frac{h(h-1)}{2}+o(1)},\qquad\text{as $\varepsilon\to 0$ %for $2\leq h\in\mathbb{N}$}, blackboard_E [ script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( italic_B ( italic_x , italic_ε ) ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ] ∼ italic_ε start_POSTSUPERSCRIPT 2 italic_h end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) start_POSTSUPERSCRIPT divide start_ARG italic_h ( italic_h - 1 ) end_ARG start_ARG 2 end_ARG + italic_o ( 1 ) end_POSTSUPERSCRIPT , as italic_ε → 0 for 2 ≤ italic_h ∈ blackboard_N , (1.15)
may suggest that the Critical 2d SHF exhibits multifractality at a logarithmic scale. This is consistent with thepicture established in [CSZ24 ] that the Critical 2d SHF just fails to be a function (it is in 𝒞 0 − superscript 𝒞 limit-from 0 {\mathcal{C}}^{0-} caligraphic_C start_POSTSUPERSCRIPT 0 - end_POSTSUPERSCRIPT ). It would be interestingto formulate the (logarithmic) multifractality features of the Critical 2d SHF. For this one would need todevelop methods complementary to those of the present articlethat would allow for asymptotics similar to (1.15 ) but for fractional moments h ∈ [ 0 , 1 ] ℎ 0 1 h\in[0,1] italic_h ∈ [ 0 , 1 ] .We conjecture that asymptotic (1.15 ) extends to h ∈ [ 0 , 1 ] ℎ 0 1 h\in[0,1] italic_h ∈ [ 0 , 1 ] (in consistency with (1.10 )).The structure of the paper is as follows. In Section 2 we recall the expression of moments of the Critical 2d SHF interms of collision diagrams as well as certain asymptotics that we will use. In Section 3 we prove the upperbound in Theorem 1.2 and in Section 4 the lower bound.
2. Auxiliary results on moments of the Critical 2d SHFIn this section we review the already established formulas of the Critical 2d SHF.The reader can find the derivation and further details at references [CSZ19b , CSZ23a , GQT21 ] .
The first moment of the Critical 2d SHF is given by
𝔼 [ 𝒵 s , t ϑ ( d x , d y ) ] = 1 2 g 1 2 ( t − s ) ( y − x ) d x d y , 𝔼 delimited-[] subscript superscript 𝒵 italic-ϑ 𝑠 𝑡
d 𝑥 d 𝑦 1 2 subscript 𝑔 1 2 𝑡 𝑠 𝑦 𝑥 d 𝑥 d 𝑦 {\mathbb{E}}[\mathscr{Z}^{\vartheta}_{s,t}(\mathrm{d}x,\mathrm{d}y)]=\tfrac{1}%{2}\,g_{\frac{1}{2}(t-s)}(y-x)\,\mathrm{d}x\,\mathrm{d}y\,, blackboard_E [ script_Z start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s , italic_t end_POSTSUBSCRIPT ( roman_d italic_x , roman_d italic_y ) ] = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_g start_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_t - italic_s ) end_POSTSUBSCRIPT ( italic_y - italic_x ) roman_d italic_x roman_d italic_y , (2.1)
where g t ( x ) = 1 2 π t e − | x | 2 2 t subscript 𝑔 𝑡 𝑥 1 2 𝜋 𝑡 superscript 𝑒 superscript 𝑥 2 2 𝑡 g_{t}(x)=\frac{1}{2\pi t}e^{-\frac{|x|^{2}}{2t}} italic_g start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_x ) = divide start_ARG 1 end_ARG start_ARG 2 italic_π italic_t end_ARG italic_e start_POSTSUPERSCRIPT - divide start_ARG | italic_x | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_t end_ARG end_POSTSUPERSCRIPT is the two-dimensional heat kernel.The covariance of the Critical 2d SHF has the expression
ℂ ov [ 𝒵 s , t ϑ ( d x , d y ) , 𝒵 s , t ϑ ( d x ′ , d y ′ ) ] ℂ ov subscript superscript 𝒵 italic-ϑ 𝑠 𝑡
d 𝑥 d 𝑦 subscript superscript 𝒵 italic-ϑ 𝑠 𝑡
d superscript 𝑥 ′ d superscript 𝑦 ′ \displaystyle\operatorname{\mathbb{C}ov}[\mathscr{Z}^{\vartheta}_{s,t}(\mathrm%{d}x,\mathrm{d}y),\mathscr{Z}^{\vartheta}_{s,t}(\mathrm{d}x^{\prime},\mathrm{d%}y^{\prime})] start_OPFUNCTION blackboard_C roman_ov end_OPFUNCTION [ script_Z start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s , italic_t end_POSTSUBSCRIPT ( roman_d italic_x , roman_d italic_y ) , script_Z start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s , italic_t end_POSTSUBSCRIPT ( roman_d italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , roman_d italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] = 1 2 K t − s ϑ ( x , x ′ ; y , y ′ ) d x d y d x ′ d y ′ , absent 1 2 superscript subscript 𝐾 𝑡 𝑠 italic-ϑ 𝑥 superscript 𝑥 ′ 𝑦 superscript 𝑦 ′ d 𝑥 d 𝑦 d superscript 𝑥 ′ d superscript 𝑦 ′ \displaystyle=\tfrac{1}{2}\,K_{t-s}^{\vartheta}(x,x^{\prime};y,y^{\prime})\,%\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}x^{\prime}\,\mathrm{d}y^{\prime}\,, = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_K start_POSTSUBSCRIPT italic_t - italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( italic_x , italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ; italic_y , italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) roman_d italic_x roman_d italic_y roman_d italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT roman_d italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , (2.2)
where
K t ϑ ( x , x ′ ; y , y ′ ) := π g t 4 ( y + y ′ 2 − x + x ′ 2 ) ∬ 0 < a < b < t g a ( x ′ − x ) G ϑ ( b − a ) g t − b ( y ′ − y ) d a d b . assign superscript subscript 𝐾 𝑡 italic-ϑ 𝑥 superscript 𝑥 ′ 𝑦 superscript 𝑦 ′ 𝜋 subscript 𝑔 𝑡 4 𝑦 superscript 𝑦 ′ 2 𝑥 superscript 𝑥 ′ 2 subscript double-integral 0 𝑎 𝑏 𝑡 subscript 𝑔 𝑎 superscript 𝑥 ′ 𝑥 subscript 𝐺 italic-ϑ 𝑏 𝑎 subscript 𝑔 𝑡 𝑏 superscript 𝑦 ′ 𝑦 differential-d 𝑎 differential-d 𝑏 \begin{split}K_{t}^{\vartheta}(x,x^{\prime};y,y^{\prime})&\,:=\,\pi\>g_{\frac{%t}{4}}\big{(}\tfrac{y+y^{\prime}}{2}-\tfrac{x+x^{\prime}}{2}\big{)}\!\!\!\iint%\limits_{0<a<b<t}\!\!\!g_{a}(x^{\prime}-x)\,G_{\vartheta}(b-a)\,g_{t-b}(y^{%\prime}-y)\,\mathrm{d}a\,\mathrm{d}b\,.\end{split} start_ROW start_CELL italic_K start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( italic_x , italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ; italic_y , italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_CELL start_CELL := italic_π italic_g start_POSTSUBSCRIPT divide start_ARG italic_t end_ARG start_ARG 4 end_ARG end_POSTSUBSCRIPT ( divide start_ARG italic_y + italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG - divide start_ARG italic_x + italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ) ∬ start_POSTSUBSCRIPT 0 < italic_a < italic_b < italic_t end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_x ) italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_b - italic_a ) italic_g start_POSTSUBSCRIPT italic_t - italic_b end_POSTSUBSCRIPT ( italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_y ) roman_d italic_a roman_d italic_b . end_CELL end_ROW (2.3)
In the above formula G ϑ ( t ) subscript 𝐺 italic-ϑ 𝑡 G_{\vartheta}(t) italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) is the density of the renewal function of the Dickman subordinator constructed in [CSZ19a ] .The exact expression of G ϑ ( t ) subscript 𝐺 italic-ϑ 𝑡 G_{\vartheta}(t) italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) is
G ϑ ( t ) = ∫ 0 ∞ e ( ϑ − γ ) s s t s − 1 Γ ( s + 1 ) d s , subscript 𝐺 italic-ϑ 𝑡 superscript subscript 0 superscript 𝑒 italic-ϑ 𝛾 𝑠 𝑠 superscript 𝑡 𝑠 1 Γ 𝑠 1 differential-d 𝑠 \displaystyle G_{\vartheta}(t)=\int_{0}^{\infty}\frac{e^{(\vartheta-\gamma)s}%st^{s-1}}{\Gamma(s+1)}\mathrm{d}s, italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT ( italic_ϑ - italic_γ ) italic_s end_POSTSUPERSCRIPT italic_s italic_t start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT end_ARG start_ARG roman_Γ ( italic_s + 1 ) end_ARG roman_d italic_s , (2.4)
where γ := − ∫ 0 ∞ log u e − u d u ≈ 0.577 … assign 𝛾 superscript subscript 0 𝑢 superscript 𝑒 𝑢 d 𝑢 0.577 … \gamma:=-\int_{0}^{\infty}\log ue^{-u}\mathrm{d}u\approx 0.577... italic_γ := - ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT roman_log italic_u italic_e start_POSTSUPERSCRIPT - italic_u end_POSTSUPERSCRIPT roman_d italic_u ≈ 0.577 … is the Euler constant and Γ ( s ) Γ 𝑠 \Gamma(s) roman_Γ ( italic_s ) is the Gamma function.For t ∈ ( 0 , 1 ) 𝑡 0 1 t\in(0,1) italic_t ∈ ( 0 , 1 ) (2.4 ) can be written as
G ϑ ( t ) = ∫ 0 ∞ e ϑ s f s ( t ) d s , subscript 𝐺 italic-ϑ 𝑡 superscript subscript 0 superscript 𝑒 italic-ϑ 𝑠 subscript 𝑓 𝑠 𝑡 differential-d 𝑠 \displaystyle G_{\vartheta}(t)=\int_{0}^{\infty}e^{\vartheta s}f_{s}(t)\,%\mathrm{d}s, italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_ϑ italic_s end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ) roman_d italic_s , (2.5)
where f s ( t ) subscript 𝑓 𝑠 𝑡 f_{s}(t) italic_f start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ) is the density of the Dickman subordinator ( Y s ) s > 0 subscript subscript 𝑌 𝑠 𝑠 0 (Y_{s})_{s>0} ( italic_Y start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_s > 0 end_POSTSUBSCRIPT – a jump process with Lévy measurex − 1 𝟙 x ∈ ( 0 , 1 ) d x superscript 𝑥 1 subscript 1 𝑥 0 1 d 𝑥 x^{-1}\mathds{1}_{x\in(0,1)}\mathrm{d}x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_x ∈ ( 0 , 1 ) end_POSTSUBSCRIPT roman_d italic_x – and is given by
f s ( t ) = { s t s − 1 e − γ s Γ ( s + 1 ) , for t ∈ ( 0 , 1 ] , s t s − 1 e − γ s Γ ( s + 1 ) − s t s − 1 ∫ 0 1 f s ( a ) ( 1 + a ) s d a , for t ∈ [ 1 , ∞ ) . subscript 𝑓 𝑠 𝑡 cases 𝑠 superscript 𝑡 𝑠 1 superscript 𝑒 𝛾 𝑠 Γ 𝑠 1 for t ∈ ( 0 , 1 ] otherwise otherwise 𝑠 superscript 𝑡 𝑠 1 superscript 𝑒 𝛾 𝑠 Γ 𝑠 1 𝑠 superscript 𝑡 𝑠 1 superscript subscript 0 1 subscript 𝑓 𝑠 𝑎 superscript 1 𝑎 𝑠 differential-d 𝑎 for t ∈ [ 1 , ∞ ) . \displaystyle f_{s}(t)=\begin{cases}\frac{st^{s-1}e^{-\gamma s}}{\Gamma(s+1)},%&\text{for $t\in(0,1]$},\\\\\frac{st^{s-1}e^{-\gamma s}}{\Gamma(s+1)}-st^{s-1}\int_{0}^{1}\frac{f_{s}(a)}{%(1+a)^{s}}\,\mathrm{d}a,&\text{for $t\in[1,\infty)$.}\end{cases} italic_f start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ) = { start_ROW start_CELL divide start_ARG italic_s italic_t start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_γ italic_s end_POSTSUPERSCRIPT end_ARG start_ARG roman_Γ ( italic_s + 1 ) end_ARG , end_CELL start_CELL for italic_t ∈ ( 0 , 1 ] , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL divide start_ARG italic_s italic_t start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_γ italic_s end_POSTSUPERSCRIPT end_ARG start_ARG roman_Γ ( italic_s + 1 ) end_ARG - italic_s italic_t start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT divide start_ARG italic_f start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_a ) end_ARG start_ARG ( 1 + italic_a ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT end_ARG roman_d italic_a , end_CELL start_CELL for italic_t ∈ [ 1 , ∞ ) . end_CELL end_ROW
The form of G ϑ ( t ) subscript 𝐺 italic-ϑ 𝑡 G_{\vartheta}(t) italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) for t ≤ 1 𝑡 1 t\leq 1 italic_t ≤ 1 , as in (2.4 ) is related to another special function, the Volterrafunction [A10 , CM24 ] . Formulas (2.2 ), (2.3 ) where first derived in the context of the Stochastic Heat Equation in [BC98 ] but the links to the Dickman subordinator were only observed later in [CSZ19a , CSZ19b ] .
The Laplace transform of (2.4 ) has a simple form, which will be useful in our analysis and so we recordit here:
Proposition 2.1. Let G ϑ ( t ) subscript 𝐺 italic-ϑ 𝑡 G_{\vartheta}(t) italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) be as in (2.4 ) for t > 0 𝑡 0 t>0 italic_t > 0 . Then for λ > e ϑ − γ 𝜆 superscript 𝑒 italic-ϑ 𝛾 \lambda>e^{\vartheta-\gamma} italic_λ > italic_e start_POSTSUPERSCRIPT italic_ϑ - italic_γ end_POSTSUPERSCRIPT we have that
∫ 0 ∞ e − λ t G ϑ ( t ) d t = 1 log λ − ϑ + γ . superscript subscript 0 superscript 𝑒 𝜆 𝑡 subscript 𝐺 italic-ϑ 𝑡 differential-d 𝑡 1 𝜆 italic-ϑ 𝛾 \displaystyle\int_{0}^{\infty}e^{-\lambda t}G_{\vartheta}(t)\,\mathrm{d}t=%\frac{1}{\log\lambda-\vartheta+\gamma}. ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ italic_t end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) roman_d italic_t = divide start_ARG 1 end_ARG start_ARG roman_log italic_λ - italic_ϑ + italic_γ end_ARG .
Proof. Replacing formula (2.4 ) into the Laplace integral and performing the integrations, we obtain:
∫ 0 ∞ G ϑ ( t ) e − λ t d t superscript subscript 0 subscript 𝐺 italic-ϑ 𝑡 superscript 𝑒 𝜆 𝑡 differential-d 𝑡 \displaystyle\int_{0}^{\infty}G_{\vartheta}(t)e^{-\lambda t}\,\mathrm{d}t ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) italic_e start_POSTSUPERSCRIPT - italic_λ italic_t end_POSTSUPERSCRIPT roman_d italic_t = ∫ 0 ∞ ∫ 0 ∞ e ( ϑ − γ ) s t s − 1 Γ ( s ) e − λ t d s d t absent superscript subscript 0 superscript subscript 0 superscript 𝑒 italic-ϑ 𝛾 𝑠 superscript 𝑡 𝑠 1 Γ 𝑠 superscript 𝑒 𝜆 𝑡 differential-d 𝑠 differential-d 𝑡 \displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\frac{e^{(\vartheta-\gamma)s}t%^{s-1}}{\Gamma(s)}e^{-\lambda t}\,\mathrm{d}s\,\mathrm{d}t = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT ( italic_ϑ - italic_γ ) italic_s end_POSTSUPERSCRIPT italic_t start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT end_ARG start_ARG roman_Γ ( italic_s ) end_ARG italic_e start_POSTSUPERSCRIPT - italic_λ italic_t end_POSTSUPERSCRIPT roman_d italic_s roman_d italic_t = ∫ 0 ∞ ( ∫ 0 ∞ t s − 1 e − λ t d t ) e ( ϑ − γ ) s Γ ( s ) d s absent superscript subscript 0 superscript subscript 0 superscript 𝑡 𝑠 1 superscript 𝑒 𝜆 𝑡 differential-d 𝑡 superscript 𝑒 italic-ϑ 𝛾 𝑠 Γ 𝑠 differential-d 𝑠 \displaystyle=\int_{0}^{\infty}\left(\int_{0}^{\infty}t^{s-1}e^{-\lambda t}%\mathrm{d}t\right)\frac{e^{(\vartheta-\gamma)s}}{\Gamma(s)}\mathrm{d}s = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_t start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ italic_t end_POSTSUPERSCRIPT roman_d italic_t ) divide start_ARG italic_e start_POSTSUPERSCRIPT ( italic_ϑ - italic_γ ) italic_s end_POSTSUPERSCRIPT end_ARG start_ARG roman_Γ ( italic_s ) end_ARG roman_d italic_s = ∫ 0 ∞ ( 1 λ s ∫ 0 ∞ t s − 1 e − t d t ) e ( ϑ − γ ) s Γ ( s ) d s absent superscript subscript 0 1 superscript 𝜆 𝑠 superscript subscript 0 superscript 𝑡 𝑠 1 superscript 𝑒 𝑡 differential-d 𝑡 superscript 𝑒 italic-ϑ 𝛾 𝑠 Γ 𝑠 differential-d 𝑠 \displaystyle=\int_{0}^{\infty}\left(\frac{1}{\lambda^{s}}\int_{0}^{\infty}t^{%s-1}e^{-t}\mathrm{d}t\right)\frac{e^{(\vartheta-\gamma)s}}{\Gamma(s)}\mathrm{d}s = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( divide start_ARG 1 end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_t start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_t end_POSTSUPERSCRIPT roman_d italic_t ) divide start_ARG italic_e start_POSTSUPERSCRIPT ( italic_ϑ - italic_γ ) italic_s end_POSTSUPERSCRIPT end_ARG start_ARG roman_Γ ( italic_s ) end_ARG roman_d italic_s = ∫ 0 ∞ 1 λ s e ( ϑ − γ ) s d s = ∫ 0 ∞ e − ( log λ − ϑ + γ ) s d s absent subscript superscript 0 1 superscript 𝜆 𝑠 superscript 𝑒 italic-ϑ 𝛾 𝑠 differential-d 𝑠 subscript superscript 0 superscript 𝑒 𝜆 italic-ϑ 𝛾 𝑠 differential-d 𝑠 \displaystyle=\int^{\infty}_{0}\frac{1}{\lambda^{s}}e^{(\vartheta-\gamma)s}%\mathrm{d}s=\int^{\infty}_{0}e^{-(\log\lambda-\vartheta+\gamma)s}\mathrm{d}s = ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT ( italic_ϑ - italic_γ ) italic_s end_POSTSUPERSCRIPT roman_d italic_s = ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - ( roman_log italic_λ - italic_ϑ + italic_γ ) italic_s end_POSTSUPERSCRIPT roman_d italic_s = 1 log λ − ϑ + γ . absent 1 𝜆 italic-ϑ 𝛾 \displaystyle=\frac{1}{\log\lambda-\vartheta+\gamma}. = divide start_ARG 1 end_ARG start_ARG roman_log italic_λ - italic_ϑ + italic_γ end_ARG .
∎
We will also need the following asymptotics for G ϑ subscript 𝐺 italic-ϑ G_{\vartheta} italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT , which were established in [CSZ19a ]
Proposition 2.2. For any ϑ ∈ ℝ italic-ϑ ℝ \vartheta\in{\mathbb{R}} italic_ϑ ∈ blackboard_R , the function G ϑ ( t ) subscript 𝐺 italic-ϑ 𝑡 G_{\vartheta}(t) italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) is continuous and strictly positive for t ∈ ( 0 , 1 ] 𝑡 0 1 t\in(0,1] italic_t ∈ ( 0 , 1 ] . As t ↓ 0 ↓ 𝑡 0 t\downarrow 0 italic_t ↓ 0 we have the asymptotic,
G ϑ ( t ) = 1 t ( log 1 t ) 2 { 1 + 2 ϑ log 1 t + O ( 1 ( log 1 t ) 2 ) } . subscript 𝐺 italic-ϑ 𝑡 1 𝑡 superscript 1 𝑡 2 1 2 italic-ϑ 1 𝑡 𝑂 1 superscript 1 𝑡 2 G_{\vartheta}(t)=\frac{1}{t(\log\frac{1}{t})^{2}}\bigg{\{}1+\frac{2\vartheta}{%\log\frac{1}{t}}+O\left(\frac{1}{(\log\frac{1}{t})^{2}}\right)\bigg{\}}. italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG 1 end_ARG start_ARG italic_t ( roman_log divide start_ARG 1 end_ARG start_ARG italic_t end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG { 1 + divide start_ARG 2 italic_ϑ end_ARG start_ARG roman_log divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_ARG + italic_O ( divide start_ARG 1 end_ARG start_ARG ( roman_log divide start_ARG 1 end_ARG start_ARG italic_t end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) } .
We next move to the formulas for higher moments. These were obtained in [CSZ19b ] in the case of the third moment andin [GQT21 ] for arbitrary moments.Here we will adopt the formulation presented in [CSZ19b ] .Let us first write the alluded formula for the h ℎ h italic_h -momentand demystify it afterwards. The formula is:
𝔼 [ ( 𝒵 t ϑ ( φ ) ) h ] = ∑ m ≥ 0 ( 2 π ) m ∑ { { i 1 , j 1 } , … , { i m , j m } ∈ { 1 , … , h } 2 with { i k , j k } ≠ { i k + 1 , j k + 1 } for k = 1 , … , m − 1 ∫ ( ℝ 2 ) h d 𝒙 ϕ ⊗ h ( 𝒙 ) \displaystyle{\mathbb{E}}\Big{[}\big{(}\mathscr{Z}_{t}^{\vartheta}(\varphi)%\big{)}^{h}\Big{]}=\sum_{m\geq 0}\,\,\,\,\,\,\,(2\pi)^{m}\hskip-34.14322pt\sum%_{\begin{subarray}{c}\{\{i_{1},j_{1}\},...,\{i_{m},j_{m}\}\in\{1,...,h\}^{2}\\\text{with $\{i_{k},j_{k}\}\neq\{i_{k+1},j_{k+1}\}$ for $k=1,...,m-1$}\end{%subarray}}\int_{(\mathbb{R}^{2})^{h}}\mathrm{d}\boldsymbol{x}\,\phi^{\otimes h%}(\boldsymbol{x}) blackboard_E [ ( script_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( italic_φ ) ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ] = ∑ start_POSTSUBSCRIPT italic_m ≥ 0 end_POSTSUBSCRIPT ( 2 italic_π ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL { { italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT } , … , { italic_i start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT } ∈ { 1 , … , italic_h } start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL with { italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } ≠ { italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT } for italic_k = 1 , … , italic_m - 1 end_CELL end_ROW end_ARG end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_d bold_italic_x italic_ϕ start_POSTSUPERSCRIPT ⊗ italic_h end_POSTSUPERSCRIPT ( bold_italic_x ) ∬ 0 ≤ a 1 < b 1 < … < a m < b m ≤ t x 1 , y 1 , … , x m , y m ∈ ℝ 2 g a 1 2 ( x 1 − x i 1 ) g a 1 2 ( x 1 − x j 1 ) ∏ r = 1 m G ϑ ( b r − a r ) g b r − a r 4 ( y r − x r ) 1 𝒮 i r , j r subscript double-integral 0 subscript 𝑎 1 subscript 𝑏 1 … subscript 𝑎 𝑚 subscript 𝑏 𝑚 𝑡 subscript 𝑥 1 subscript 𝑦 1 … subscript 𝑥 𝑚 subscript 𝑦 𝑚
superscript ℝ 2
subscript 𝑔 subscript 𝑎 1 2 subscript 𝑥 1 superscript 𝑥 subscript 𝑖 1 subscript 𝑔 subscript 𝑎 1 2 subscript 𝑥 1 superscript 𝑥 subscript 𝑗 1 superscript subscript product 𝑟 1 𝑚 subscript 𝐺 italic-ϑ subscript 𝑏 𝑟 subscript 𝑎 𝑟 subscript 𝑔 subscript 𝑏 𝑟 subscript 𝑎 𝑟 4 subscript 𝑦 𝑟 subscript 𝑥 𝑟 subscript 1 subscript 𝒮 subscript 𝑖 𝑟 subscript 𝑗 𝑟
\displaystyle\iint_{\begin{subarray}{c}0\leq a_{1}<b_{1}<...<a_{m}<b_{m}\leq t%\\x_{1},y_{1},...,x_{m},y_{m}\in\mathbb{R}^{2}\end{subarray}}g_{\frac{a_{1}}{2}}%(x_{1}-x^{i_{1}})g_{\frac{a_{1}}{2}}(x_{1}-x^{j_{1}})\,\,\prod_{r=1}^{m}G_{%\vartheta}(b_{r}-a_{r})g_{\frac{b_{r}-a_{r}}{4}}(y_{r}-x_{r})\,\,\mathds{1}_{{%\mathcal{S}}_{i_{r},j_{r}}} ∬ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL 0 ≤ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < … < italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT < italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≤ italic_t end_CELL end_ROW start_ROW start_CELL italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_x start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_x start_POSTSUPERSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) ∏ start_POSTSUBSCRIPT italic_r = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) italic_g start_POSTSUBSCRIPT divide start_ARG italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) blackboard_1 start_POSTSUBSCRIPT caligraphic_S start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT × ( ∏ 1 ≤ r ≤ m − 1 g a r + 1 − b 𝗉 ( i r + 1 ) 2 ( x r + 1 − y 𝗉 ( i r + 1 ) ) g a r + 1 − b 𝗉 ( j r + 1 ) 2 ( x r + 1 − y 𝗉 ( j r + 1 ) ) ) d 𝒙 → d 𝒚 → d a → d b → absent subscript product 1 𝑟 𝑚 1 subscript 𝑔 subscript 𝑎 𝑟 1 subscript 𝑏 𝗉 subscript 𝑖 𝑟 1 2 subscript 𝑥 𝑟 1 subscript 𝑦 𝗉 subscript 𝑖 𝑟 1 subscript 𝑔 subscript 𝑎 𝑟 1 subscript 𝑏 𝗉 subscript 𝑗 𝑟 1 2 subscript 𝑥 𝑟 1 subscript 𝑦 𝗉 subscript 𝑗 𝑟 1 d → 𝒙 d → 𝒚 d → 𝑎 d → 𝑏 \displaystyle\,\,\times\Big{(}\prod_{1\leq r\leq m-1}g_{\frac{a_{r+1}-b_{%\mathsf{p}(i_{r+1})}}{2}}(x_{r+1}-y_{\mathsf{p}(i_{r+1})})\,g_{\frac{a_{r+1}-b%_{\mathsf{p}(j_{r+1})}}{2}}(x_{r+1}-y_{\mathsf{p}(j_{r+1})})\Big{)}\mathrm{d}%\vec{\boldsymbol{x}}\,\mathrm{d}\vec{\boldsymbol{y}}\,\mathrm{d}\vec{a}\,%\mathrm{d}\vec{b}\,\, × ( ∏ start_POSTSUBSCRIPT 1 ≤ italic_r ≤ italic_m - 1 end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) ) roman_d over→ start_ARG bold_italic_x end_ARG roman_d over→ start_ARG bold_italic_y end_ARG roman_d over→ start_ARG italic_a end_ARG roman_d over→ start_ARG italic_b end_ARG (2.6)
where ϕ ⊗ h ( 𝒙 ) := ϕ ( x 1 ) ⋯ ϕ ( x h ) assign superscript italic-ϕ tensor-product absent ℎ 𝒙 italic-ϕ superscript 𝑥 1 ⋯ italic-ϕ superscript 𝑥 ℎ \phi^{\otimes h}(\boldsymbol{x}):=\phi(x^{1})\cdots\phi(x^{h}) italic_ϕ start_POSTSUPERSCRIPT ⊗ italic_h end_POSTSUPERSCRIPT ( bold_italic_x ) := italic_ϕ ( italic_x start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ⋯ italic_ϕ ( italic_x start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ) ,𝒮 i r , j r subscript 𝒮 subscript 𝑖 𝑟 subscript 𝑗 𝑟
{\mathcal{S}}_{i_{r},j_{r}} caligraphic_S start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT is the event that Brownian motions i r subscript 𝑖 𝑟 i_{r} italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and j r subscript 𝑗 𝑟 j_{r} italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , only, are involved in collisions in the time interval( a r , b r ) subscript 𝑎 𝑟 subscript 𝑏 𝑟 (a_{r},b_{r}) ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) conditioned to both start at positions x r subscript 𝑥 𝑟 x_{r} italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and ending at positions y r subscript 𝑦 𝑟 y_{r} italic_y start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and for a pair { i r , j r } subscript 𝑖 𝑟 subscript 𝑗 𝑟 \{i_{r},j_{r}\} { italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT } we define
𝗉 ( i r ) 𝗉 subscript 𝑖 𝑟 \displaystyle\mathsf{p}(i_{r}) sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) := i ℓ ( r ) with ℓ ( r ) := max { 0 ≤ ℓ < r : 𝟙 𝒮 i ℓ , j ℓ = 1 and i r ∈ { i ℓ , j ℓ } } formulae-sequence assign absent subscript 𝑖 ℓ 𝑟 with
assign ℓ 𝑟 : 0 ℓ 𝑟 subscript 1 subscript 𝒮 subscript 𝑖 ℓ subscript 𝑗 ℓ
1 and subscript 𝑖 𝑟 subscript 𝑖 ℓ subscript 𝑗 ℓ \displaystyle:=i_{\ell(r)}\quad\text{with}\quad\ell(r):=\max\big{\{}0\leq\ell<%r\colon\mathds{1}_{{\mathcal{S}}_{i_{\ell},j_{\ell}}}=1\,\,\text{and}\,\,i_{r}%\in\{i_{\ell},j_{\ell}\}\big{\}} := italic_i start_POSTSUBSCRIPT roman_ℓ ( italic_r ) end_POSTSUBSCRIPT with roman_ℓ ( italic_r ) := roman_max { 0 ≤ roman_ℓ < italic_r : blackboard_1 start_POSTSUBSCRIPT caligraphic_S start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 1 and italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ∈ { italic_i start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT } }
and similarly for 𝗉 ( j r ) 𝗉 subscript 𝑗 𝑟 \mathsf{p}(j_{r}) sansserif_p ( italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) .In other words, 𝗉 ( i r ) 𝗉 subscript 𝑖 𝑟 \mathsf{p}(i_{r}) sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) is the last time before r 𝑟 r italic_r that Brownian motion B ( i r ) superscript 𝐵 subscript 𝑖 𝑟 B^{(i_{r})} italic_B start_POSTSUPERSCRIPT ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT was involved in a collision. We note that if 𝗉 ( i r ) = 0 𝗉 subscript 𝑖 𝑟 0 \mathsf{p}(i_{r})=0 sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = 0 then ( b 𝗉 ( i r ) , y 𝗉 ( i r ) ) := ( 0 , x i r ) assign subscript 𝑏 𝗉 subscript 𝑖 𝑟 subscript 𝑦 𝗉 subscript 𝑖 𝑟 0 superscript 𝑥 subscript 𝑖 𝑟 (b_{\mathsf{p}(i_{r})},y_{\mathsf{p}(i_{r})}):=(0,x^{i_{r}}) ( italic_b start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) := ( 0 , italic_x start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) .A diagrammatic representation of formula (2 ) is shown in Figure 1 .To get a better idea of formula (2 ) and its diagrammatic representation,we may use the Feynman-Kac formula (1.2 ) from which an easy computationgives that
𝔼 [ ( ∫ ℝ 2 ϕ ( x ) u ε ( t , x ) d x ) h ] = ∫ ( ℝ 2 ) h ϕ ⊗ h ( 𝒙 ) 𝐄 𝒙 ⊗ h [ ( β ε 2 ∑ 1 ≤ i < j ≤ h ∫ 0 t J ε ( B s ( i ) − B s ( j ) ) d s ) ] d 𝒙 𝔼 delimited-[] superscript subscript superscript ℝ 2 italic-ϕ 𝑥 superscript 𝑢 𝜀 𝑡 𝑥 differential-d 𝑥 ℎ subscript superscript superscript ℝ 2 ℎ superscript italic-ϕ tensor-product absent ℎ 𝒙 superscript subscript 𝐄 𝒙 tensor-product absent ℎ delimited-[] superscript subscript 𝛽 𝜀 2 subscript 1 𝑖 𝑗 ℎ superscript subscript 0 𝑡 subscript 𝐽 𝜀 superscript subscript 𝐵 𝑠 𝑖 superscript subscript 𝐵 𝑠 𝑗 differential-d 𝑠 differential-d 𝒙 \displaystyle{\mathbb{E}}\Big{[}\Big{(}\int_{\mathbb{R}^{2}}\phi(x)u^{%\varepsilon}(t,x)\mathrm{d}x\Big{)}^{h}\Big{]}=\int_{(\mathbb{R}^{2})^{h}}\phi%^{\otimes h}(\boldsymbol{x})\,\boldsymbol{\mathrm{E}}_{\boldsymbol{x}}^{%\otimes h}\Big{[}\Big{(}\beta_{\varepsilon}^{2}\sum_{1\leq i<j\leq h}\int_{0}^%{t}J_{\varepsilon}(B_{s}^{(i)}-B_{s}^{(j)})\,\mathrm{d}s\Big{)}\Big{]}\,%\mathrm{d}\boldsymbol{x} blackboard_E [ ( ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ϕ ( italic_x ) italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_t , italic_x ) roman_d italic_x ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ] = ∫ start_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ϕ start_POSTSUPERSCRIPT ⊗ italic_h end_POSTSUPERSCRIPT ( bold_italic_x ) bold_E start_POSTSUBSCRIPT bold_italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊗ italic_h end_POSTSUPERSCRIPT [ ( italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT 1 ≤ italic_i < italic_j ≤ italic_h end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_J start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ( italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT - italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ) roman_d italic_s ) ] roman_d bold_italic_x (2.7)
with 𝒙 = ( x 1 , … , x h ) 𝒙 superscript 𝑥 1 … superscript 𝑥 ℎ \boldsymbol{x}=(x^{1},...,x^{h}) bold_italic_x = ( italic_x start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_x start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ) . J ε ( x ) := β ε 2 1 ε 2 J ( x ε ) assign subscript 𝐽 𝜀 𝑥 superscript subscript 𝛽 𝜀 2 1 superscript 𝜀 2 𝐽 𝑥 𝜀 J_{\varepsilon}(x):=\beta_{\varepsilon}^{2}\,\frac{1}{\varepsilon^{2}}J\big{(}%\frac{x}{\varepsilon}\big{)} italic_J start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ( italic_x ) := italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_J ( divide start_ARG italic_x end_ARG start_ARG italic_ε end_ARG ) ,with J = j ∗ j 𝐽 𝑗 𝑗 J=j*j italic_J = italic_j ∗ italic_j and j 𝑗 j italic_j as in (1.2 ),approximates a delta function when ε → 0 → 𝜀 0 \varepsilon\to 0 italic_ε → 0 . When β ε subscript 𝛽 𝜀 \beta_{\varepsilon} italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT is chosenat the critical value (1.5 ), then the main contribution to (2.7 ), in the limit ε → 0 → 𝜀 0 \varepsilon\to 0 italic_ε → 0 comes from configurations where the Brownian motions B ( 1 ) , … , B ( h ) superscript 𝐵 1 … superscript 𝐵 ℎ
B^{(1)},...,B^{(h)} italic_B start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , … , italic_B start_POSTSUPERSCRIPT ( italic_h ) end_POSTSUPERSCRIPT have pairwise collisions.Expanding the exponential in (2.7 ) and breaking down according to when and where the collisions take placeand which Brownian motions are involved, it gives rise to formula (2 )and its graphical representation as depicted in Figure 1 . The wiggle lines appearing in thatFigure represent the weights accumulated from collisions of the Brownian motions and we often call itreplica overlap .
Our main objective, which will be carried in the next sections, is to determine the asymptotics of(2 ) when the test function ϕ italic-ϕ \phi italic_ϕ is 𝒰 B ( 0 , ε ) ( ⋅ ) := 1 π ε 2 1 B ( 0 , ε ) ( ⋅ ) assign subscript 𝒰 𝐵 0 𝜀 ⋅ 1 𝜋 superscript 𝜀 2 subscript 1 𝐵 0 𝜀 ⋅ {\mathcal{U}}_{B(0,\varepsilon)}(\cdot):=\frac{1}{\pi\varepsilon^{2}}\,\mathds%{1}_{B(0,\varepsilon)}(\cdot) caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ( ⋅ ) := divide start_ARG 1 end_ARG start_ARG italic_π italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG blackboard_1 start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ( ⋅ ) .However, it will be more convenient to work with ϕ italic-ϕ \phi italic_ϕ being a heat kernel approximation of the delta functionand look into the asymptotics of
𝒵 ε ϑ , h := 𝔼 [ ( 𝒵 1 ϑ ( g ε 2 2 ) ) h ] with g ε 2 2 ( x ) = 1 π ε 2 e − | x | 2 ε 2 , formulae-sequence assign superscript subscript 𝒵 𝜀 italic-ϑ ℎ
𝔼 delimited-[] superscript superscript subscript 𝒵 1 italic-ϑ subscript 𝑔 superscript 𝜀 2 2 ℎ with
subscript 𝑔 superscript 𝜀 2 2 𝑥 1 𝜋 superscript 𝜀 2 superscript 𝑒 superscript 𝑥 2 superscript 𝜀 2 \displaystyle\mathscr{Z}_{\varepsilon}^{\vartheta,h}:={\mathbb{E}}\Big{[}\big{%(}\mathscr{Z}_{1}^{\vartheta}(g_{\frac{\varepsilon^{2}}{2}})\big{)}^{h}\Big{]}%\qquad\text{with}\qquad g_{\frac{\varepsilon^{2}}{2}}(x)=\frac{1}{\pi%\varepsilon^{2}}e^{-\frac{|x|^{2}}{\varepsilon^{2}}}, script_Z start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT := blackboard_E [ ( script_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( italic_g start_POSTSUBSCRIPT divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ] with italic_g start_POSTSUBSCRIPT divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x ) = divide start_ARG 1 end_ARG start_ARG italic_π italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - divide start_ARG | italic_x | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_POSTSUPERSCRIPT , (2.8)
and then perform a comparison to 𝔼 [ ( 𝒵 1 ϑ ( 𝒰 B ( 0 , ε ) ) ) h ] 𝔼 delimited-[] superscript superscript subscript 𝒵 1 italic-ϑ subscript 𝒰 𝐵 0 𝜀 ℎ {\mathbb{E}}\Big{[}\big{(}\mathscr{Z}_{1}^{\vartheta}({\mathcal{U}}_{B(0,%\varepsilon)})\big{)}^{h}\Big{]} blackboard_E [ ( script_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT ( caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ] .For simplicity we just consider time t = 1 𝑡 1 t=1 italic_t = 1 .
Let us write the series expression for 𝒵 ε ϑ , h superscript subscript 𝒵 𝜀 italic-ϑ ℎ
\mathscr{Z}_{\varepsilon}^{\vartheta,h} script_Z start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT .For every i 𝑖 i italic_i , we integrate,g ε 2 2 ( x i ) subscript 𝑔 superscript 𝜀 2 2 superscript 𝑥 𝑖 g_{\frac{\varepsilon^{2}}{2}}(x^{i}) italic_g start_POSTSUBSCRIPT divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) against the heat kernel corresponding to the weight of the laceemanating from x i superscript 𝑥 𝑖 x^{i} italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT (see Figure 1 ):
∫ ℝ 2 g ε 2 2 ( x i ) g a 𝗋 ( i ) 2 ( x 𝗋 ( i ) − x i ) d x i = g a 𝗋 ( i ) + ε 2 2 ( y 𝗋 ( i ) − x i ) , subscript superscript ℝ 2 subscript 𝑔 superscript 𝜀 2 2 superscript 𝑥 𝑖 subscript 𝑔 subscript 𝑎 𝗋 𝑖 2 subscript 𝑥 𝗋 𝑖 superscript 𝑥 𝑖 differential-d superscript 𝑥 𝑖 subscript 𝑔 subscript 𝑎 𝗋 𝑖 superscript 𝜀 2 2 subscript 𝑦 𝗋 𝑖 superscript 𝑥 𝑖 \displaystyle\int_{\mathbb{R}^{2}}g_{\frac{\varepsilon^{2}}{2}}(x^{i})\,g_{%\frac{a_{\mathsf{r}(i)}}{2}}(x_{\mathsf{r}(i)}-x^{i})\,\mathrm{d}x^{i}=g_{%\frac{a_{\mathsf{r}(i)}+\varepsilon^{2}}{2}}(y_{\mathsf{r}(i)}-x^{i}), ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT sansserif_r ( italic_i ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT sansserif_r ( italic_i ) end_POSTSUBSCRIPT - italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) roman_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT sansserif_r ( italic_i ) end_POSTSUBSCRIPT + italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT sansserif_r ( italic_i ) end_POSTSUBSCRIPT - italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) ,
where we have denoted by 𝗋 ( i ) 𝗋 𝑖 \mathsf{r}(i) sansserif_r ( italic_i ) the index which determines the point ( a r , x r ) , r = 1 , … , m formulae-sequence subscript 𝑎 𝑟 subscript 𝑥 𝑟 𝑟
1 … 𝑚
(a_{r},x_{r}),r=1,...,m ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) , italic_r = 1 , … , italic_m that is connected to ( 0 , x i ) 0 superscript 𝑥 𝑖 (0,x^{i}) ( 0 , italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) . Performing all such integrations over the initial points x i , i = 1 , … , h formulae-sequence superscript 𝑥 𝑖 𝑖
1 … ℎ
x^{i},i=1,...,h italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_i = 1 , … , italic_h and shifting the time variables a 1 , b 1 , … , a r , b r subscript 𝑎 1 subscript 𝑏 1 … subscript 𝑎 𝑟 subscript 𝑏 𝑟
a_{1},b_{1},...,a_{r},b_{r} italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT by ε 2 superscript 𝜀 2 \varepsilon^{2} italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , we arrive at the following formula,which is depicted in Figure 2 :
𝒵 ε ϑ , h = ∑ m ≥ 0 ( 2 π ) m ∑ { { i 1 , j 1 } , … , { i m , j m } ∈ { 1 , … , h } 2 with { i k , j k } ≠ { i k + 1 , j k + 1 } for k = 1 , … , m − 1 \displaystyle\mathscr{Z}_{\varepsilon}^{\vartheta,h}=\sum_{m\geq 0}\,\,\,\,\,%\,\,(2\pi)^{m}\hskip-34.14322pt\sum_{\begin{subarray}{c}\{\{i_{1},j_{1}\},...,%\{i_{m},j_{m}\}\in\{1,...,h\}^{2}\\\text{with $\{i_{k},j_{k}\}\neq\{i_{k+1},j_{k+1}\}$ for $k=1,...,m-1$}\end{%subarray}} script_Z start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_m ≥ 0 end_POSTSUBSCRIPT ( 2 italic_π ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL { { italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT } , … , { italic_i start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT } ∈ { 1 , … , italic_h } start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL with { italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } ≠ { italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT } for italic_k = 1 , … , italic_m - 1 end_CELL end_ROW end_ARG end_POSTSUBSCRIPT ∬ ε 2 ≤ a 1 < b 1 < … < a m < b m ≤ 1 + ε 2 x 1 , y 1 , … , x m , y m ∈ ℝ 2 g a 1 2 ( x 1 ) 2 ∏ r = 1 m G ϑ ( b r − a r ) g b r − a r 4 ( y r − x r ) 1 𝒮 i r , j r subscript double-integral superscript 𝜀 2 subscript 𝑎 1 subscript 𝑏 1 … subscript 𝑎 𝑚 subscript 𝑏 𝑚 1 superscript 𝜀 2 subscript 𝑥 1 subscript 𝑦 1 … subscript 𝑥 𝑚 subscript 𝑦 𝑚
superscript ℝ 2
subscript 𝑔 subscript 𝑎 1 2 superscript subscript 𝑥 1 2 superscript subscript product 𝑟 1 𝑚 subscript 𝐺 italic-ϑ subscript 𝑏 𝑟 subscript 𝑎 𝑟 subscript 𝑔 subscript 𝑏 𝑟 subscript 𝑎 𝑟 4 subscript 𝑦 𝑟 subscript 𝑥 𝑟 subscript 1 subscript 𝒮 subscript 𝑖 𝑟 subscript 𝑗 𝑟
\displaystyle\iint_{\begin{subarray}{c}\varepsilon^{2}\leq a_{1}<b_{1}<...<a_{%m}<b_{m}\leq 1+\varepsilon^{2}\\x_{1},y_{1},...,x_{m},y_{m}\in\mathbb{R}^{2}\end{subarray}}g_{\frac{a_{1}}{2}}%(x_{1})^{2}\,\,\prod_{r=1}^{m}G_{\vartheta}(b_{r}-a_{r})g_{\frac{b_{r}-a_{r}}{%4}}(y_{r}-x_{r})\,\,\mathds{1}_{{\mathcal{S}}_{i_{r},j_{r}}} ∬ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < … < italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT < italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≤ 1 + italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∏ start_POSTSUBSCRIPT italic_r = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) italic_g start_POSTSUBSCRIPT divide start_ARG italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) blackboard_1 start_POSTSUBSCRIPT caligraphic_S start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT (2.9) × ( ∏ 1 ≤ r ≤ m − 1 g a r + 1 − b 𝗉 ( i r + 1 ) 2 ( x r + 1 − y 𝗉 ( i r + 1 ) ) g a r + 1 − b 𝗉 ( j r + 1 ) 2 ( x r + 1 − y 𝗉 ( j r + 1 ) ) ) d 𝒙 → d 𝒚 → d a → d b → , absent subscript product 1 𝑟 𝑚 1 subscript 𝑔 subscript 𝑎 𝑟 1 subscript 𝑏 𝗉 subscript 𝑖 𝑟 1 2 subscript 𝑥 𝑟 1 subscript 𝑦 𝗉 subscript 𝑖 𝑟 1 subscript 𝑔 subscript 𝑎 𝑟 1 subscript 𝑏 𝗉 subscript 𝑗 𝑟 1 2 subscript 𝑥 𝑟 1 subscript 𝑦 𝗉 subscript 𝑗 𝑟 1 d → 𝒙 d → 𝒚 d → 𝑎 d → 𝑏 \displaystyle\times\Big{(}\prod_{1\leq r\leq m-1}g_{\frac{a_{r+1}-b_{\mathsf{p%}(i_{r+1})}}{2}}(x_{r+1}-y_{\mathsf{p}(i_{r+1})})\,g_{\frac{a_{r+1}-b_{\mathsf%{p}(j_{r+1})}}{2}}(x_{r+1}-y_{\mathsf{p}(j_{r+1})})\Big{)}\,\,\mathrm{d}\vec{%\boldsymbol{x}}\,\mathrm{d}\vec{\boldsymbol{y}}\,\mathrm{d}\vec{a}\,\mathrm{d}%\vec{b}\,\,\,, × ( ∏ start_POSTSUBSCRIPT 1 ≤ italic_r ≤ italic_m - 1 end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) ) roman_d over→ start_ARG bold_italic_x end_ARG roman_d over→ start_ARG bold_italic_y end_ARG roman_d over→ start_ARG italic_a end_ARG roman_d over→ start_ARG italic_b end_ARG ,
We note that if 𝗉 ( i r ) = 0 𝗉 subscript 𝑖 𝑟 0 \mathsf{p}(i_{r})=0 sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = 0 , then ( b 𝗉 ( i r ) , y 𝗉 ( i r ) ) = ( 0 , 0 ) subscript 𝑏 𝗉 subscript 𝑖 𝑟 subscript 𝑦 𝗉 subscript 𝑖 𝑟 0 0 (b_{\mathsf{p}(i_{r})},y_{\mathsf{p}(i_{r})})=(0,0) ( italic_b start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) = ( 0 , 0 ) .
3. Upper boundIn this section we prove the upper bound in Theorem 1.2 . The main estimate is contained inthe following proposition:
Proposition 3.1. Recall the definition of 𝒵 ε ϑ , h superscript subscript 𝒵 𝜀 italic-ϑ ℎ
\mathscr{Z}_{\varepsilon}^{\vartheta,h} script_Z start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT from (2.8 ).For any δ > 0 𝛿 0 \delta>0 italic_δ > 0 , h ≥ 2 ℎ 2 h\geq 2 italic_h ≥ 2 and ϑ ∈ ℝ italic-ϑ ℝ \vartheta\in{\mathbb{R}} italic_ϑ ∈ blackboard_R , then
𝒵 ε ϑ , h ≤ ( log 1 ε ) ( h 2 ) + o ( 1 ) , as ε → 0 . subscript superscript 𝒵 italic-ϑ ℎ
𝜀 superscript 1 𝜀 binomial ℎ 2 𝑜 1 as ε → 0
\displaystyle\mathscr{Z}^{\vartheta,h}_{\varepsilon}\leq\left(\log\frac{1}{%\varepsilon}\right)^{{h\choose 2}+o(1)},\qquad\text{as $\varepsilon\rightarrow0%$}. script_Z start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ≤ ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) start_POSTSUPERSCRIPT ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) + italic_o ( 1 ) end_POSTSUPERSCRIPT , as italic_ε → 0 . (3.1)
Having the above estimate at hand we can deducethe upper bound in (1.13 ) as follows:
Proof of the upper bound in Theorem 1.2 . We have the comparison:
𝒰 B ( 0 , ε ) ( ⋅ ) = 1 π ε 2 𝟙 B ( 0 , ε ) ( ⋅ ) ≤ e g ε 2 / 2 ( ⋅ ) . subscript 𝒰 𝐵 0 𝜀 ⋅ 1 𝜋 superscript 𝜀 2 subscript 1 𝐵 0 𝜀 ⋅ 𝑒 subscript 𝑔 superscript 𝜀 2 2 ⋅ \displaystyle{\mathcal{U}}_{B(0,\varepsilon)}(\cdot)=\frac{1}{\pi\varepsilon^{%2}}\mathds{1}_{B(0,\varepsilon)}(\cdot)\leq eg_{\varepsilon^{2}/2}(\cdot). caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ( ⋅ ) = divide start_ARG 1 end_ARG start_ARG italic_π italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG blackboard_1 start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ( ⋅ ) ≤ italic_e italic_g start_POSTSUBSCRIPT italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 end_POSTSUBSCRIPT ( ⋅ ) .
Hence, by Proposition 3.1 ,
𝔼 [ ( 𝒵 1 ϑ ( 𝒰 B ( 0 , ε ) ) ) h ] ≤ e h 𝒵 ε ϑ , h ≤ e h ( log 1 ε ) ( h 2 ) + o ( 1 ) . 𝔼 delimited-[] superscript subscript superscript 𝒵 italic-ϑ 1 subscript 𝒰 𝐵 0 𝜀 ℎ superscript 𝑒 ℎ subscript superscript 𝒵 italic-ϑ ℎ
𝜀 superscript 𝑒 ℎ superscript 1 𝜀 binomial ℎ 2 𝑜 1 \displaystyle{\mathbb{E}}\bigg{[}\left(\mathscr{Z}^{\vartheta}_{1}\left({%\mathcal{U}}_{B(0,\varepsilon)}\right)\right)^{h}\bigg{]}\leq e^{h}\mathscr{Z}%^{\vartheta,h}_{\varepsilon}\leq e^{h}\left(\log\frac{1}{\varepsilon}\right)^{%{h\choose 2}+o(1)}. blackboard_E [ ( script_Z start_POSTSUPERSCRIPT italic_ϑ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ] ≤ italic_e start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT script_Z start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ≤ italic_e start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) start_POSTSUPERSCRIPT ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) + italic_o ( 1 ) end_POSTSUPERSCRIPT .
as ε → 0 → 𝜀 0 \varepsilon\rightarrow 0 italic_ε → 0 .∎
The rest of the section is devoted to the proof of Proposition 3.1 .As a warm up computation, we start with the following preliminary estimate on𝒵 ε ϑ , h superscript subscript 𝒵 𝜀 italic-ϑ ℎ
\mathscr{Z}_{\varepsilon}^{\vartheta,h} script_Z start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT :
Lemma 3.2. For 0 < ε < 1 0 𝜀 1 0<\varepsilon<1 0 < italic_ε < 1 , the following estimate holds:
𝒵 ε ϑ , h subscript superscript 𝒵 italic-ϑ ℎ
𝜀 \displaystyle\mathscr{Z}^{\vartheta,h}_{\varepsilon} script_Z start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ≤ ∑ m ≥ 0 ℐ m , h , ε absent subscript 𝑚 0 subscript ℐ 𝑚 ℎ 𝜀
\displaystyle\leq\sum_{m\geq 0}\mathscr{I}_{m,h,\varepsilon} ≤ ∑ start_POSTSUBSCRIPT italic_m ≥ 0 end_POSTSUBSCRIPT script_I start_POSTSUBSCRIPT italic_m , italic_h , italic_ε end_POSTSUBSCRIPT (3.2)
where ℐ 0 , h , ε = 1 subscript ℐ 0 ℎ 𝜀
1 \mathscr{I}_{0,h,\varepsilon}=1 script_I start_POSTSUBSCRIPT 0 , italic_h , italic_ε end_POSTSUBSCRIPT = 1 , ℐ 1 , h , ε = C ( h 2 ) log 1 ε subscript ℐ 1 ℎ 𝜀
𝐶 binomial ℎ 2 1 𝜀 \mathscr{I}_{1,h,\varepsilon}=C{h\choose 2}\log\frac{1}{\varepsilon} script_I start_POSTSUBSCRIPT 1 , italic_h , italic_ε end_POSTSUBSCRIPT = italic_C ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG for some C > 0 𝐶 0 C>0 italic_C > 0 , and for m ≥ 2 𝑚 2 m\geq 2 italic_m ≥ 2 :
ℐ m , h , ε := ( h 2 ) [ ( h 2 ) − 1 ] m − 1 ∫⋯∫ ∑ i ( u i + v i ) ≤ 1 + ε 2 , u 1 > ε 2 1 u 1 ∏ 1 ≤ r ≤ m − 1 G ϑ ( v r ) 1 2 ( v r + u r ) + u r + 1 G ϑ ( v m ) d u → d v → assign subscript ℐ 𝑚 ℎ 𝜀
binomial ℎ 2 superscript delimited-[] binomial ℎ 2 1 𝑚 1 subscript multiple-integral formulae-sequence subscript 𝑖 subscript 𝑢 𝑖 subscript 𝑣 𝑖 1 superscript 𝜀 2 subscript 𝑢 1 superscript 𝜀 2 1 subscript 𝑢 1 subscript product 1 𝑟 𝑚 1 subscript 𝐺 italic-ϑ subscript 𝑣 𝑟 1 2 subscript 𝑣 𝑟 subscript 𝑢 𝑟 subscript 𝑢 𝑟 1 subscript 𝐺 italic-ϑ subscript 𝑣 𝑚 d → 𝑢 d → 𝑣 \displaystyle\mathscr{I}_{m,h,\varepsilon}:={h\choose 2}\left[{h\choose 2}-1%\right]^{m-1}\hskip-14.22636pt\idotsint\limits_{\sum_{i}(u_{i}+v_{i})\leq 1+%\varepsilon^{2}\,,\,u_{1}>\varepsilon^{2}}\hskip-2.84544pt\frac{1}{u_{1}}\prod%_{1\leq r\leq m-1}\frac{G_{\vartheta}(v_{r})}{\frac{1}{2}(v_{r}+u_{r})+u_{r+1}%}\,G_{\vartheta}(v_{m})\,\mathrm{d}\vec{u}\,\mathrm{d}\vec{v} script_I start_POSTSUBSCRIPT italic_m , italic_h , italic_ε end_POSTSUBSCRIPT := ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) [ ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) - 1 ] start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT ∫⋯∫ start_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≤ 1 + italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ∏ start_POSTSUBSCRIPT 1 ≤ italic_r ≤ italic_m - 1 end_POSTSUBSCRIPT divide start_ARG italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) end_ARG start_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_v start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) + italic_u start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT end_ARG italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) roman_d over→ start_ARG italic_u end_ARG roman_d over→ start_ARG italic_v end_ARG (3.3)
Proof. We work with ( 2 ).The m = 0 𝑚 0 m=0 italic_m = 0 term in that formula is simply 1 1 1 1 . The m = 1 𝑚 1 m=1 italic_m = 1 term is equal to:
2 π ∑ i , j ∈ { 1 , … , h } ∬ x , y ∈ ℝ 2 ε 2 ≤ a < b ≤ 1 + ε 2 g a 2 ( x ) 2 G ϑ ( b − a ) g b − a 4 ( y − x ) d x d y d a d b . 2 𝜋 subscript 𝑖 𝑗
1 … ℎ subscript double-integral superscript 𝑥 𝑦
superscript ℝ 2 superscript 𝜀 2 𝑎 𝑏 1 superscript 𝜀 2 subscript 𝑔 𝑎 2 superscript 𝑥 2 subscript 𝐺 italic-ϑ 𝑏 𝑎 subscript 𝑔 𝑏 𝑎 4 𝑦 𝑥 d 𝑥 d 𝑦 d 𝑎 d 𝑏
\displaystyle 2\pi\sum_{i,j\in\{1,...,h\}}\iint_{\stackrel{{\scriptstyle%\varepsilon^{2}\leq a<b\leq 1+\varepsilon^{2}}}{{x,y\in{\mathbb{R}}^{2}}}}%\quad g_{\frac{a}{2}}(x)^{2}G_{\vartheta}(b-a)g_{\frac{b-a}{4}}(y-x)\,\mathrm{%d}x\,\mathrm{d}y\,\mathrm{d}a\,\mathrm{d}b. 2 italic_π ∑ start_POSTSUBSCRIPT italic_i , italic_j ∈ { 1 , … , italic_h } end_POSTSUBSCRIPT ∬ start_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG italic_x , italic_y ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_a < italic_b ≤ 1 + italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_RELOP end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_a end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_b - italic_a ) italic_g start_POSTSUBSCRIPT divide start_ARG italic_b - italic_a end_ARG start_ARG 4 end_ARG end_POSTSUBSCRIPT ( italic_y - italic_x ) roman_d italic_x roman_d italic_y roman_d italic_a roman_d italic_b . (3.4)
To simplify notations, we extend the integral ∬ ε 2 ≤ a < b ≤ 1 + ε 2 ( … ) d a d b subscript double-integral superscript 𝜀 2 𝑎 𝑏 1 superscript 𝜀 2 … differential-d 𝑎 differential-d 𝑏 \iint_{\varepsilon^{2}\leq a<b\leq 1+\varepsilon^{2}}(...)\mathrm{d}a\mathrm{d}b ∬ start_POSTSUBSCRIPT italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_a < italic_b ≤ 1 + italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( … ) roman_d italic_a roman_d italic_b to ∬ ε 2 ≤ a < b ≤ 2 ( … ) d a d b subscript double-integral superscript 𝜀 2 𝑎 𝑏 2 … differential-d 𝑎 differential-d 𝑏 \iint_{\varepsilon^{2}\leq a<b\leq 2}(...)\mathrm{d}a\mathrm{d}b ∬ start_POSTSUBSCRIPT italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_a < italic_b ≤ 2 end_POSTSUBSCRIPT ( … ) roman_d italic_a roman_d italic_b . We first perform the integration over y 𝑦 y italic_y , which gives
∫ ℝ 2 g b − a 4 ( y − x ) d y = 1 . subscript superscript ℝ 2 subscript 𝑔 𝑏 𝑎 4 𝑦 𝑥 differential-d 𝑦 1 \displaystyle\int_{{\mathbb{R}}^{2}}g_{\frac{b-a}{4}}(y-x)\mathrm{d}y=1. ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_b - italic_a end_ARG start_ARG 4 end_ARG end_POSTSUBSCRIPT ( italic_y - italic_x ) roman_d italic_y = 1 .
Then by Proposition 2.2 , we integrate over b 𝑏 b italic_b :
∫ a 2 G ϑ ( b − a ) d b ≤ C . superscript subscript 𝑎 2 subscript 𝐺 italic-ϑ 𝑏 𝑎 differential-d 𝑏 𝐶 \displaystyle\int_{a}^{2}G_{\vartheta}(b-a)\mathrm{d}b\leq C. ∫ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_b - italic_a ) roman_d italic_b ≤ italic_C .
Therefore we bound ( 3.4 ) by:
C ∑ i , j ∈ { 1 , … , h } ∬ ε 2 ≤ a ≤ 2 , x ∈ ℝ 2 g a 2 ( x ) 2 d x d a 𝐶 subscript 𝑖 𝑗
1 … ℎ subscript double-integral formulae-sequence superscript 𝜀 2 𝑎 2 𝑥 superscript ℝ 2 subscript 𝑔 𝑎 2 superscript 𝑥 2 d 𝑥 d 𝑎
\displaystyle C\sum_{i,j\in\{1,...,h\}}\iint_{\varepsilon^{2}\leq a\leq 2\,,x%\in{\mathbb{R}}^{2}}\quad g_{\frac{a}{2}}(x)^{2}\,\mathrm{d}x\mathrm{d}a italic_C ∑ start_POSTSUBSCRIPT italic_i , italic_j ∈ { 1 , … , italic_h } end_POSTSUBSCRIPT ∬ start_POSTSUBSCRIPT italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_a ≤ 2 , italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_a end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_d italic_x roman_d italic_a = C ( h 2 ) ∬ ε 2 ≤ a ≤ 2 , x ∈ ℝ 2 g a 2 ( x ) 2 d x d a absent 𝐶 binomial ℎ 2 subscript double-integral formulae-sequence superscript 𝜀 2 𝑎 2 𝑥 superscript ℝ 2 subscript 𝑔 𝑎 2 superscript 𝑥 2 differential-d 𝑥 differential-d 𝑎 \displaystyle=C{h\choose 2}\iint_{\varepsilon^{2}\leq a\leq 2\,,x\in{\mathbb{R%}}^{2}}g_{\frac{a}{2}}(x)^{2}\,\mathrm{d}x\mathrm{d}a = italic_C ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) ∬ start_POSTSUBSCRIPT italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_a ≤ 2 , italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_a end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_d italic_x roman_d italic_a = C ( h 2 ) ∫ ε 2 2 g a ( 0 ) d a absent 𝐶 binomial ℎ 2 superscript subscript superscript 𝜀 2 2 subscript 𝑔 𝑎 0 differential-d 𝑎 \displaystyle=C{h\choose 2}\int_{\varepsilon^{2}}^{2}g_{a}(0)\mathrm{d}a = italic_C ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) ∫ start_POSTSUBSCRIPT italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( 0 ) roman_d italic_a = C ( h 2 ) log ( 2 ε 2 ) ≤ C ( h 2 ) log ( 1 ε ) . absent 𝐶 binomial ℎ 2 2 superscript 𝜀 2 𝐶 binomial ℎ 2 1 𝜀 \displaystyle=C{h\choose 2}\log(\frac{2}{\varepsilon^{2}})\leq C{h\choose 2}%\log(\frac{1}{\varepsilon}). = italic_C ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) roman_log ( divide start_ARG 2 end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ≤ italic_C ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) .
Now we treat the case m ≥ 2 𝑚 2 m\geq 2 italic_m ≥ 2 . We will follow the convention that b 0 = 0 subscript 𝑏 0 0 b_{0}=0 italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0 . and recall the convention that if 𝗉 ( i r ) = 0 𝗉 subscript 𝑖 𝑟 0 \mathsf{p}(i_{r})=0 sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = 0 , then ( b 𝗉 ( i r ) , y 𝗉 ( i r ) ) = ( 0 , 0 ) subscript 𝑏 𝗉 subscript 𝑖 𝑟 subscript 𝑦 𝗉 subscript 𝑖 𝑟 0 0 (b_{\mathsf{p}(i_{r})},y_{\mathsf{p}(i_{r})})=(0,0) ( italic_b start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) = ( 0 , 0 ) .We start by performing the integration over y m subscript 𝑦 𝑚 y_{m} italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , which amounts to
∫ ℝ 2 g b m − a m 4 ( y m − x m ) d y m = 1 . subscript superscript ℝ 2 subscript 𝑔 subscript 𝑏 𝑚 subscript 𝑎 𝑚 4 subscript 𝑦 𝑚 subscript 𝑥 𝑚 differential-d subscript 𝑦 𝑚 1 \displaystyle\int_{\mathbb{R}^{2}}g_{\frac{b_{m}-a_{m}}{4}}(y_{m}-x_{m})\,%\mathrm{d}y_{m}=1. ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) roman_d italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 1 .
Next we integrate x m subscript 𝑥 𝑚 x_{m} italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT .
∫ ℝ 2 g a m − b ℐ 𝗆 , 𝗁 , ε 𝗉 ( 𝗂 𝗆 ) 2 ( x m − y 𝗉 ( i m ) ) g a m − b 𝗉 ( j m ) 2 ( x m − y 𝗉 ( j m ) ) d x m subscript superscript ℝ 2 subscript 𝑔 subscript 𝑎 𝑚 subscript 𝑏 subscript ℐ 𝗆 𝗁 𝜀
𝗉 subscript 𝗂 𝗆 2 subscript 𝑥 𝑚 subscript 𝑦 𝗉 subscript 𝑖 𝑚 subscript 𝑔 subscript 𝑎 𝑚 subscript 𝑏 𝗉 subscript 𝑗 𝑚 2 subscript 𝑥 𝑚 subscript 𝑦 𝗉 subscript 𝑗 𝑚 differential-d subscript 𝑥 𝑚 \displaystyle\int_{{\mathbb{R}}^{2}}g_{\frac{a_{m}-b_{\sf\mathscr{I}_{m,h,%\varepsilon}p(i_{m})}}{2}}\big{(}x_{m}-y_{\mathsf{p}(i_{m})}\big{)}\,g_{\frac{%a_{m}-b_{\mathsf{p}(j_{m})}}{2}}\big{(}x_{m}-y_{\mathsf{p}(j_{m})}\big{)}\,%\mathrm{d}x_{m} ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT script_I start_POSTSUBSCRIPT sansserif_m , sansserif_h , italic_ε end_POSTSUBSCRIPT sansserif_p ( sansserif_i start_POSTSUBSCRIPT sansserif_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) italic_g start_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) roman_d italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = g a m − 1 2 ( b 𝗉 ( i m ) + b 𝗉 ( j m ) ) ( y 𝗉 ( i m ) − y 𝗉 ( j m ) ) ≤ 1 / π 2 a m − ( b 𝗉 ( i m ) + b 𝗉 ( j m ) ) ≤ 1 / π ( a m − b m − 1 ) + ( a m − b m − 2 ) , absent subscript 𝑔 subscript 𝑎 𝑚 1 2 subscript 𝑏 𝗉 subscript 𝑖 𝑚 subscript 𝑏 𝗉 subscript 𝑗 𝑚 subscript 𝑦 𝗉 subscript 𝑖 𝑚 subscript 𝑦 𝗉 subscript 𝑗 𝑚 1 𝜋 2 subscript 𝑎 𝑚 subscript 𝑏 𝗉 subscript 𝑖 𝑚 subscript 𝑏 𝗉 subscript 𝑗 𝑚 1 𝜋 subscript 𝑎 𝑚 subscript 𝑏 𝑚 1 subscript 𝑎 𝑚 subscript 𝑏 𝑚 2 \displaystyle=g_{a_{m}-\frac{1}{2}(b_{\mathsf{p}(i_{m})}+b_{\mathsf{p}(j_{m})}%)}\big{(}y_{\mathsf{p}(i_{m})}-y_{\mathsf{p}(j_{m})}\big{)}\leq\frac{1/\pi}{2a%_{m}-(b_{\mathsf{p}(i_{m})}\!+b_{\mathsf{p}(j_{m})})}\leq\frac{1/\pi}{(a_{m}-b%_{m-1})+(a_{m}-b_{m-2})}, = italic_g start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_b start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) ≤ divide start_ARG 1 / italic_π end_ARG start_ARG 2 italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - ( italic_b start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) end_ARG ≤ divide start_ARG 1 / italic_π end_ARG start_ARG ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ) + ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT italic_m - 2 end_POSTSUBSCRIPT ) end_ARG ,
as b 𝗉 ( i m ) subscript 𝑏 𝗉 subscript 𝑖 𝑚 b_{\mathsf{p}(i_{m})} italic_b start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT and b 𝗉 ( j m ) subscript 𝑏 𝗉 subscript 𝑗 𝑚 b_{\mathsf{p}(j_{m})} italic_b start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT may be before b m − 1 subscript 𝑏 𝑚 1 b_{m-1} italic_b start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT and b m − 2 subscript 𝑏 𝑚 2 b_{m-2} italic_b start_POSTSUBSCRIPT italic_m - 2 end_POSTSUBSCRIPT , respectively, but notafter and they cannot be both equal to just one of b m − 1 subscript 𝑏 𝑚 1 b_{m-1} italic_b start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT or b m − 2 subscript 𝑏 𝑚 2 b_{m-2} italic_b start_POSTSUBSCRIPT italic_m - 2 end_POSTSUBSCRIPT .
The result then follows by iterating the same integration successively over y m − 1 , x m − 1 , … , y 1 , x 1 subscript 𝑦 𝑚 1 subscript 𝑥 𝑚 1 … subscript 𝑦 1 subscript 𝑥 1
y_{m-1},x_{m-1},...,y_{1},x_{1} italic_y start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and changing variables as
v i := b i − a i and u i := a i − b i − 1 . formulae-sequence assign subscript 𝑣 𝑖 subscript 𝑏 𝑖 subscript 𝑎 𝑖 and
assign subscript 𝑢 𝑖 subscript 𝑎 𝑖 subscript 𝑏 𝑖 1 \displaystyle v_{i}:=b_{i}-a_{i}\qquad\text{and}\qquad u_{i}:=a_{i}-b_{i-1}. italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT .
The combinatorial factor ( h 2 ) [ ( h 2 ) − 1 ] m − 1 binomial ℎ 2 superscript delimited-[] binomial ℎ 2 1 𝑚 1 {h\choose 2}\left[{h\choose 2}-1\right]^{m-1} ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) [ ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) - 1 ] start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT counts the choices ofassigning pairs { i , j } 𝑖 𝑗 \{i,j\} { italic_i , italic_j } to the wiggle lines, noting that two consecutive wiggle lines will need tohave different pairs assigned to them.∎
We will next bound (3.2 ). The first step is to introduce multipliers and integrateover the v 1 , … , v r subscript 𝑣 1 … subscript 𝑣 𝑟
v_{1},...,v_{r} italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT variables to obtain the following intermediate estimate:
Lemma 3.3 (Integration of the replica variables). There exists a constant C > 0 𝐶 0 C>0 italic_C > 0 such that for all λ > e ϑ − γ 𝜆 superscript 𝑒 italic-ϑ 𝛾 \lambda>e^{\vartheta-\gamma} italic_λ > italic_e start_POSTSUPERSCRIPT italic_ϑ - italic_γ end_POSTSUPERSCRIPT , it holds
𝒵 ε ϑ , h ≤ C e 2 λ ∑ m = 0 ∞ ℐ m , h , ε ( λ ) subscript superscript 𝒵 italic-ϑ ℎ
𝜀 𝐶 superscript 𝑒 2 𝜆 superscript subscript 𝑚 0 subscript superscript ℐ 𝜆 𝑚 ℎ 𝜀
\mathscr{Z}^{\vartheta,h}_{\varepsilon}\leq Ce^{2\lambda}\sum_{m=0}^{\infty}%\mathscr{I}^{(\lambda)}_{m,h,\varepsilon} script_Z start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ≤ italic_C italic_e start_POSTSUPERSCRIPT 2 italic_λ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT script_I start_POSTSUPERSCRIPT ( italic_λ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_h , italic_ε end_POSTSUBSCRIPT (3.5)
where ℐ m , h , ε ( λ ) := ℐ m , h , ε assign subscript superscript ℐ 𝜆 𝑚 ℎ 𝜀
subscript ℐ 𝑚 ℎ 𝜀
\mathscr{I}^{(\lambda)}_{m,h,\varepsilon}:=\mathscr{I}_{m,h,\varepsilon} script_I start_POSTSUPERSCRIPT ( italic_λ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_h , italic_ε end_POSTSUBSCRIPT := script_I start_POSTSUBSCRIPT italic_m , italic_h , italic_ε end_POSTSUBSCRIPT for m = 0 , 1 𝑚 0 1
m=0,1 italic_m = 0 , 1 , and for m ≥ 2 𝑚 2 m\geq 2 italic_m ≥ 2 :
ℐ m , h , ε ( λ ) := ( h 2 ) [ ( h 2 ) − 1 ] m − 1 ∫⋯∫ ∑ i u i ≤ 2 , u 1 > ε 2 1 u 1 ∏ r = 2 m F λ ( u r + u r − 1 2 ) d u → , assign subscript superscript ℐ 𝜆 𝑚 ℎ 𝜀
binomial ℎ 2 superscript delimited-[] binomial ℎ 2 1 𝑚 1 subscript multiple-integral formulae-sequence subscript 𝑖 subscript 𝑢 𝑖 2 subscript 𝑢 1 superscript 𝜀 2 1 subscript 𝑢 1 subscript superscript product 𝑚 𝑟 2 subscript 𝐹 𝜆 subscript 𝑢 𝑟 subscript 𝑢 𝑟 1 2 d → 𝑢 \mathscr{I}^{(\lambda)}_{m,h,\varepsilon}:={h\choose 2}\left[{h\choose 2}-1%\right]^{m-1}\idotsint\limits_{\sum_{i}u_{i}\leq 2\,,\,u_{1}>\varepsilon^{2}}%\frac{1}{u_{1}}\prod^{m}_{r=2}F_{\lambda}\big{(}u_{r}+\frac{u_{r-1}}{2}\big{)}%\,\,\mathrm{d}\vec{u}, script_I start_POSTSUPERSCRIPT ( italic_λ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_h , italic_ε end_POSTSUBSCRIPT := ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) [ ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) - 1 ] start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT ∫⋯∫ start_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 2 , italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ∏ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r = 2 end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + divide start_ARG italic_u start_POSTSUBSCRIPT italic_r - 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) roman_d over→ start_ARG italic_u end_ARG , (3.6)
with
F λ ( w ) := ∫ 0 ∞ e − σ w d σ log ( λ + σ / 2 ) − ϑ + γ . assign subscript 𝐹 𝜆 𝑤 subscript superscript 0 superscript 𝑒 𝜎 𝑤 d 𝜎 𝜆 𝜎 2 italic-ϑ 𝛾 F_{\lambda}(w):=\int^{\infty}_{0}\frac{e^{-\sigma w}\,\mathrm{d}\sigma}{\log(%\lambda+\sigma/2)-\vartheta+\gamma}. italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_w ) := ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_σ italic_w end_POSTSUPERSCRIPT roman_d italic_σ end_ARG start_ARG roman_log ( italic_λ + italic_σ / 2 ) - italic_ϑ + italic_γ end_ARG . (3.7)
Proof. For m < 2 𝑚 2 m<2 italic_m < 2 there is nothing to prove. For m ≥ 2 𝑚 2 m\geq 2 italic_m ≥ 2 , to simplify notationally, we extend the integral in ( 3.2 ) to ∑ i ( u i + v i ) < 2 subscript 𝑖 subscript 𝑢 𝑖 subscript 𝑣 𝑖 2 \sum_{i}(u_{i}+v_{i})<2 ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) < 2 .We next introduce the multipliers.To this end, we consider a parameter λ > 0 𝜆 0 \lambda>0 italic_λ > 0 , which will be suitably chosen later onand we multiply ( 3.2 ) by e 2 λ e − λ ∑ i v i ≥ 1 superscript 𝑒 2 𝜆 superscript 𝑒 𝜆 subscript 𝑖 subscript 𝑣 𝑖 1 e^{2\lambda}e^{-\lambda\sum_{i}v_{i}}\geq 1 italic_e start_POSTSUPERSCRIPT 2 italic_λ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ≥ 1 to obtain the bound
ℐ m , h , ε ≤ e 2 λ ( h 2 ) [ ( h 2 ) − 1 ] m − 1 ∫⋯∫ ∑ i ( u i + v i ) ≤ 2 , u 1 > ε 2 1 u 1 ∏ 1 ≤ r ≤ m − 1 e − λ v r G ϑ ( v r ) 1 2 ( v r + u r ) + u r + 1 G ϑ ( v m ) d u → d v → . subscript ℐ 𝑚 ℎ 𝜀
superscript 𝑒 2 𝜆 binomial ℎ 2 superscript delimited-[] binomial ℎ 2 1 𝑚 1 subscript multiple-integral formulae-sequence subscript 𝑖 subscript 𝑢 𝑖 subscript 𝑣 𝑖 2 subscript 𝑢 1 superscript 𝜀 2 1 subscript 𝑢 1 subscript product 1 𝑟 𝑚 1 superscript 𝑒 𝜆 subscript 𝑣 𝑟 subscript 𝐺 italic-ϑ subscript 𝑣 𝑟 1 2 subscript 𝑣 𝑟 subscript 𝑢 𝑟 subscript 𝑢 𝑟 1 subscript 𝐺 italic-ϑ subscript 𝑣 𝑚 d → 𝑢 d → 𝑣 \begin{split}\mathscr{I}_{m,h,\varepsilon}&\leq e^{2\lambda}{h\choose 2}\left[%{h\choose 2}-1\right]^{m-1}\hskip-14.22636pt\idotsint\limits_{\sum_{i}(u_{i}+v%_{i})\leq 2\,,\,u_{1}>\varepsilon^{2}}\hskip-14.22636pt\frac{1}{u_{1}}\prod_{1%\leq r\leq m-1}\frac{e^{-\lambda v_{r}}\,G_{\vartheta}(v_{r})}{\frac{1}{2}(v_{%r}+u_{r})+u_{r+1}}\,G_{\vartheta}(v_{m})\,\,\mathrm{d}\vec{u}\mathrm{d}\vec{v}%.\end{split} start_ROW start_CELL script_I start_POSTSUBSCRIPT italic_m , italic_h , italic_ε end_POSTSUBSCRIPT end_CELL start_CELL ≤ italic_e start_POSTSUPERSCRIPT 2 italic_λ end_POSTSUPERSCRIPT ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) [ ( binomial start_ARG italic_h end_ARG start_ARG 2 end_ARG ) - 1 ] start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT ∫⋯∫ start_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≤ 2 , italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ∏ start_POSTSUBSCRIPT 1 ≤ italic_r ≤ italic_m - 1 end_POSTSUBSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_λ italic_v start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) end_ARG start_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_v start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) + italic_u start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT end_ARG italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) roman_d over→ start_ARG italic_u end_ARG roman_d over→ start_ARG italic_v end_ARG . end_CELL end_ROW (3.8)
Next we integrate all the v 𝑣 v italic_v variables. Starting from v m subscript 𝑣 𝑚 v_{m} italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT we use the bound
∫ 0 2 G ϑ ( v m ) e − λ v m d v m ≤ ∫ 0 2 G ϑ ( v m ) d v m ≤ C superscript subscript 0 2 subscript 𝐺 italic-ϑ subscript 𝑣 𝑚 superscript 𝑒 𝜆 subscript 𝑣 𝑚 differential-d subscript 𝑣 𝑚 superscript subscript 0 2 subscript 𝐺 italic-ϑ subscript 𝑣 𝑚 differential-d subscript 𝑣 𝑚 𝐶 \int_{0}^{2}G_{\vartheta}(v_{m})e^{-\lambda v_{m}}\mathrm{d}v_{m}\leq\int_{0}^%{2}G_{\vartheta}(v_{m})\mathrm{d}v_{m}\leq C ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_λ italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_d italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≤ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) roman_d italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≤ italic_C (3.9)
which follows from Proposition 2.2 . For the rest of the v 𝑣 v italic_v -variables we usethat for any w > 0 𝑤 0 w>0 italic_w > 0 we have the bound:
∫ 0 2 e − λ v G ϑ ( v ) v / 2 + w d v superscript subscript 0 2 superscript 𝑒 𝜆 𝑣 subscript 𝐺 italic-ϑ 𝑣 𝑣 2 𝑤 differential-d 𝑣 \displaystyle\int_{0}^{2}\frac{e^{-\lambda v}\,G_{\vartheta}(v)}{v/2+w}\,%\mathrm{d}v ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_λ italic_v end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_v ) end_ARG start_ARG italic_v / 2 + italic_w end_ARG roman_d italic_v = ∫ 0 2 ∫ 0 ∞ e − σ ( v / 2 + w ) e − λ v G ϑ ( v ) d σ d v absent subscript superscript 2 0 subscript superscript 0 superscript 𝑒 𝜎 𝑣 2 𝑤 superscript 𝑒 𝜆 𝑣 subscript 𝐺 italic-ϑ 𝑣 differential-d 𝜎 differential-d 𝑣 \displaystyle=\int^{2}_{0}\int^{\infty}_{0}e^{-\sigma(v/2+w)}e^{-\lambda v}G_{%\vartheta}(v)\,\mathrm{d}\sigma\,\mathrm{d}v = ∫ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - italic_σ ( italic_v / 2 + italic_w ) end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ italic_v end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_v ) roman_d italic_σ roman_d italic_v ≤ ∫ 0 ∞ d σ e − σ w ∫ 0 ∞ e − ( λ + σ / 2 ) v G ϑ ( v ) d v absent subscript superscript 0 differential-d 𝜎 superscript 𝑒 𝜎 𝑤 subscript superscript 0 superscript 𝑒 𝜆 𝜎 2 𝑣 subscript 𝐺 italic-ϑ 𝑣 differential-d 𝑣 \displaystyle\leq\int^{\infty}_{0}\mathrm{d}\sigma\,e^{-\sigma w}\int^{\infty}%_{0}e^{-(\lambda+\sigma/2)v}\,G_{\vartheta}(v)\mathrm{d}v ≤ ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_d italic_σ italic_e start_POSTSUPERSCRIPT - italic_σ italic_w end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - ( italic_λ + italic_σ / 2 ) italic_v end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_v ) roman_d italic_v = ∫ 0 ∞ e − σ w d σ log ( λ + σ / 2 ) − ϑ + γ , absent subscript superscript 0 superscript 𝑒 𝜎 𝑤 d 𝜎 𝜆 𝜎 2 italic-ϑ 𝛾 \displaystyle=\int^{\infty}_{0}\frac{e^{-\sigma w}\,\mathrm{d}\sigma}{\log(%\lambda+\sigma/2)-\vartheta+\gamma}, = ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_σ italic_w end_POSTSUPERSCRIPT roman_d italic_σ end_ARG start_ARG roman_log ( italic_λ + italic_σ / 2 ) - italic_ϑ + italic_γ end_ARG , (3.10)
where in the last step we used Proposition 2.1 . For the last formula to be valid we need,according to Proposition 2.1 , to choose λ > e ϑ − γ 𝜆 superscript 𝑒 italic-ϑ 𝛾 \lambda>e^{\vartheta-\gamma} italic_λ > italic_e start_POSTSUPERSCRIPT italic_ϑ - italic_γ end_POSTSUPERSCRIPT .To conclude, we choose w := u r + 1 2 u r − 1 assign 𝑤 subscript 𝑢 𝑟 1 2 subscript 𝑢 𝑟 1 w:=u_{r}+\frac{1}{2}u_{r-1} italic_w := italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_u start_POSTSUBSCRIPT italic_r - 1 end_POSTSUBSCRIPT and insert successively for r = 2 , … , m 𝑟 2 … 𝑚
r=2,...,m italic_r = 2 , … , italic_m .∎
The next step is to integrate over the u 𝑢 u italic_u variables in (3.6 ). Our approach here isinspired by [CZ23 ] .However, some details are rather different as we make use of the multiplier λ 𝜆 \lambda italic_λ andwe also take into account the specifics of the criticality of the Critical 2d SHF.
To start with we define:
f λ ( w ) := ∫ w 2 F λ ( v ) d v = ∫ 0 ∞ 1 σ e − σ w − e − 2 σ log ( λ + σ / 2 ) − ( ϑ − γ ) d σ . assign subscript 𝑓 𝜆 𝑤 subscript superscript 2 𝑤 subscript 𝐹 𝜆 𝑣 differential-d 𝑣 subscript superscript 0 1 𝜎 superscript 𝑒 𝜎 𝑤 superscript 𝑒 2 𝜎 𝜆 𝜎 2 italic-ϑ 𝛾 differential-d 𝜎 f_{\lambda}(w):=\int^{2}_{w}F_{\lambda}(v)\mathrm{d}v=\int^{\infty}_{0}\frac{1%}{\sigma}\frac{e^{-\sigma w}-e^{-2\sigma}}{\log(\lambda+\sigma/2)-(\vartheta-%\gamma)}\,\mathrm{d}\sigma. italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_w ) := ∫ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_v ) roman_d italic_v = ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_σ end_ARG divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_σ italic_w end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT - 2 italic_σ end_POSTSUPERSCRIPT end_ARG start_ARG roman_log ( italic_λ + italic_σ / 2 ) - ( italic_ϑ - italic_γ ) end_ARG roman_d italic_σ . (3.11)
Note that f λ ′ = − F λ ≤ 0 subscript superscript 𝑓 ′ 𝜆 subscript 𝐹 𝜆 0 f^{\prime}_{\lambda}=-F_{\lambda}\leq 0 italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT = - italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ≤ 0 on ( 0 , 2 ] 0 2 (0,2] ( 0 , 2 ] , as F 𝐹 F italic_F is non-negative, thus, f 𝑓 f italic_f is non-increasing.We have the following Lemma:
Lemma 3.4. There exists C > 0 𝐶 0 C>0 italic_C > 0 such that for all λ > ( e 2 ( ϑ − γ ) ∨ 1 ) 𝜆 superscript 𝑒 2 italic-ϑ 𝛾 1 \lambda>\left(e^{2(\vartheta-\gamma)}\vee 1\right) italic_λ > ( italic_e start_POSTSUPERSCRIPT 2 ( italic_ϑ - italic_γ ) end_POSTSUPERSCRIPT ∨ 1 ) and w ∈ ( 0 , 1 ) 𝑤 0 1 w\in(0,1) italic_w ∈ ( 0 , 1 ) we have:
∫ 0 2 F λ ( u + w ) f λ ( u ) j d u ≤ ∑ ℓ = 0 j + 1 j ! ( j + 1 − ℓ ) ! ( 4 log λ ) ℓ f λ ( 2 w ) j + 1 − ℓ . superscript subscript 0 2 subscript 𝐹 𝜆 𝑢 𝑤 subscript 𝑓 𝜆 superscript 𝑢 𝑗 differential-d 𝑢 subscript superscript 𝑗 1 ℓ 0 𝑗 𝑗 1 ℓ superscript 4 𝜆 ℓ subscript 𝑓 𝜆 superscript 2 𝑤 𝑗 1 ℓ \displaystyle\int_{0}^{2}F_{\lambda}(u+w)f_{\lambda}(u)^{j}\,\mathrm{d}u\leq%\sum^{j+1}_{\ell=0}\frac{j!}{(j+1-\ell)!}\Big{(}\frac{4}{\log\lambda}\Big{)}^{%\ell}f_{\lambda}(2w)^{j+1-\ell}. ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u + italic_w ) italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT roman_d italic_u ≤ ∑ start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ = 0 end_POSTSUBSCRIPT divide start_ARG italic_j ! end_ARG start_ARG ( italic_j + 1 - roman_ℓ ) ! end_ARG ( divide start_ARG 4 end_ARG start_ARG roman_log italic_λ end_ARG ) start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( 2 italic_w ) start_POSTSUPERSCRIPT italic_j + 1 - roman_ℓ end_POSTSUPERSCRIPT . (3.12)
Proof. We start using the monotonicity of F λ subscript 𝐹 𝜆 F_{\lambda} italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT and noting that for λ > e 2 ( ϑ − γ ) 𝜆 superscript 𝑒 2 italic-ϑ 𝛾 \lambda>\text{$e^{2(\vartheta-\gamma)}$} italic_λ > italic_e start_POSTSUPERSCRIPT 2 ( italic_ϑ - italic_γ ) end_POSTSUPERSCRIPT and u ≥ 0 𝑢 0 u\geq 0 italic_u ≥ 0 :
F λ ( u + w ) subscript 𝐹 𝜆 𝑢 𝑤 \displaystyle F_{\lambda}(u+w) italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u + italic_w ) ≤ F λ ( w ) = ∫ 0 ∞ e − σ w d σ log ( λ + σ / 2 ) − ( ϑ − γ ) ≤ 2 ∫ 0 ∞ e − σ w log λ d σ = 2 w log λ . absent subscript 𝐹 𝜆 𝑤 subscript superscript 0 superscript 𝑒 𝜎 𝑤 d 𝜎 𝜆 𝜎 2 italic-ϑ 𝛾 2 subscript superscript 0 superscript 𝑒 𝜎 𝑤 𝜆 differential-d 𝜎 2 𝑤 𝜆 \displaystyle\leq F_{\lambda}(w)=\int^{\infty}_{0}\frac{e^{-\sigma w}\,\mathrm%{d}\sigma}{\log(\lambda+\sigma/2)-(\vartheta-\gamma)}\leq 2\int^{\infty}_{0}%\frac{e^{-\sigma w}}{\log\lambda}\mathrm{d}\sigma=\frac{2}{w\log\lambda}. ≤ italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_w ) = ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_σ italic_w end_POSTSUPERSCRIPT roman_d italic_σ end_ARG start_ARG roman_log ( italic_λ + italic_σ / 2 ) - ( italic_ϑ - italic_γ ) end_ARG ≤ 2 ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_σ italic_w end_POSTSUPERSCRIPT end_ARG start_ARG roman_log italic_λ end_ARG roman_d italic_σ = divide start_ARG 2 end_ARG start_ARG italic_w roman_log italic_λ end_ARG . (3.13)
We next split the integral on the left-hand side of ( 3.12 ) into ∫ 0 2 w ( ⋯ ) d u superscript subscript 0 2 𝑤 ⋯ differential-d 𝑢 \int_{0}^{2w}(\cdots)\mathrm{d}u ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 italic_w end_POSTSUPERSCRIPT ( ⋯ ) roman_d italic_u and ∫ 2 w 2 ( ⋯ ) d u superscript subscript 2 𝑤 2 ⋯ differential-d 𝑢 \int_{2w}^{2}(\cdots)\mathrm{d}u ∫ start_POSTSUBSCRIPT 2 italic_w end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( ⋯ ) roman_d italic_u , which we call I 𝐼 I italic_I and I I 𝐼 𝐼 II italic_I italic_I , respectively.We start by estimating integral I 𝐼 I italic_I . By ( 3.13 ) we have,
I = ∫ 0 2 w F λ ( u + w ) f λ ( u ) j d u ≤ 2 w log λ ∫ 0 2 w f λ ( u ) j d u 𝐼 subscript superscript 2 𝑤 0 subscript 𝐹 𝜆 𝑢 𝑤 subscript 𝑓 𝜆 superscript 𝑢 𝑗 differential-d 𝑢 2 𝑤 𝜆 subscript superscript 2 𝑤 0 subscript 𝑓 𝜆 superscript 𝑢 𝑗 differential-d 𝑢 \begin{split}I&=\int^{2w}_{0}F_{\lambda}(u+w)f_{\lambda}(u)^{j}\mathrm{d}u\leq%\frac{2}{w\log\lambda}\int^{2w}_{0}f_{\lambda}(u)^{j}\mathrm{d}u\\\end{split} start_ROW start_CELL italic_I end_CELL start_CELL = ∫ start_POSTSUPERSCRIPT 2 italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u + italic_w ) italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT roman_d italic_u ≤ divide start_ARG 2 end_ARG start_ARG italic_w roman_log italic_λ end_ARG ∫ start_POSTSUPERSCRIPT 2 italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT roman_d italic_u end_CELL end_ROW (3.14)
By integration by parts we have,
∫ 0 2 w f λ ( u ) j 𝑑 u = 2 w f λ ( 2 w ) j − j ∫ 0 2 w u f λ ′ ( u ) f λ ( u ) j − 1 d u ≤ 2 w f λ ( 2 w ) j + j 2 log λ ∫ 0 2 w f λ ( u ) j − 1 d u , subscript superscript 2 𝑤 0 subscript 𝑓 𝜆 superscript 𝑢 𝑗 differential-d 𝑢 2 𝑤 subscript 𝑓 𝜆 superscript 2 𝑤 𝑗 𝑗 subscript superscript 2 𝑤 0 𝑢 subscript superscript 𝑓 ′ 𝜆 𝑢 subscript 𝑓 𝜆 superscript 𝑢 𝑗 1 differential-d 𝑢 2 𝑤 subscript 𝑓 𝜆 superscript 2 𝑤 𝑗 𝑗 2 𝜆 subscript superscript 2 𝑤 0 subscript 𝑓 𝜆 superscript 𝑢 𝑗 1 differential-d 𝑢 \begin{split}\int^{2w}_{0}f_{\lambda}(u)^{j}du&=2wf_{\lambda}(2w)^{j}-j\int^{2%w}_{0}uf^{\prime}_{\lambda}(u)f_{\lambda}(u)^{j-1}\mathrm{d}u\\&\leq 2wf_{\lambda}(2w)^{j}+j\frac{2}{\log\lambda}\int^{2w}_{0}f_{\lambda}(u)^%{j-1}\mathrm{d}u,\end{split} start_ROW start_CELL ∫ start_POSTSUPERSCRIPT 2 italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_d italic_u end_CELL start_CELL = 2 italic_w italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( 2 italic_w ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - italic_j ∫ start_POSTSUPERSCRIPT 2 italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_u italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT roman_d italic_u end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ 2 italic_w italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( 2 italic_w ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT + italic_j divide start_ARG 2 end_ARG start_ARG roman_log italic_λ end_ARG ∫ start_POSTSUPERSCRIPT 2 italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT roman_d italic_u , end_CELL end_ROW
where in the inequality we used ( 3.13 ) and − u f λ ( u ) ′ = u F λ ( u ) ≤ 2 log λ 𝑢 subscript 𝑓 𝜆 superscript 𝑢 ′ 𝑢 subscript 𝐹 𝜆 𝑢 2 𝜆 -uf_{\lambda}(u)^{\prime}=uF_{\lambda}(u)\leq\frac{2}{\log\lambda} - italic_u italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_u italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) ≤ divide start_ARG 2 end_ARG start_ARG roman_log italic_λ end_ARG . Iterating this computation we have that, for j ≥ 1 𝑗 1 j\geq 1 italic_j ≥ 1 ,
∫ 0 2 w f λ ( u ) j d u ≤ 2 w ∑ i = 0 j j ! ( j − i ) ! ( 2 log λ ) i f λ ( 2 w ) j − i , subscript superscript 2 𝑤 0 subscript 𝑓 𝜆 superscript 𝑢 𝑗 differential-d 𝑢 2 𝑤 subscript superscript 𝑗 𝑖 0 𝑗 𝑗 𝑖 superscript 2 𝜆 𝑖 subscript 𝑓 𝜆 superscript 2 𝑤 𝑗 𝑖 \displaystyle\int^{2w}_{0}f_{\lambda}(u)^{j}\mathrm{d}u\leq 2w\sum^{j}_{i=0}%\frac{j!}{(j-i)!}\Big{(}\frac{2}{\log\lambda}\Big{)}^{i}f_{\lambda}(2w)^{j-i}, ∫ start_POSTSUPERSCRIPT 2 italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT roman_d italic_u ≤ 2 italic_w ∑ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT divide start_ARG italic_j ! end_ARG start_ARG ( italic_j - italic_i ) ! end_ARG ( divide start_ARG 2 end_ARG start_ARG roman_log italic_λ end_ARG ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( 2 italic_w ) start_POSTSUPERSCRIPT italic_j - italic_i end_POSTSUPERSCRIPT ,
and so
I ≤ ∑ i = 0 j j ! ( j − i ) ! ( 4 log λ ) i + 1 f λ ( 2 w ) j − i . 𝐼 subscript superscript 𝑗 𝑖 0 𝑗 𝑗 𝑖 superscript 4 𝜆 𝑖 1 subscript 𝑓 𝜆 superscript 2 𝑤 𝑗 𝑖 \displaystyle I\leq\sum^{j}_{i=0}\frac{j!}{(j-i)!}\Big{(}\frac{4}{\log\lambda}%\Big{)}^{i+1}f_{\lambda}(2w)^{j-i}. italic_I ≤ ∑ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT divide start_ARG italic_j ! end_ARG start_ARG ( italic_j - italic_i ) ! end_ARG ( divide start_ARG 4 end_ARG start_ARG roman_log italic_λ end_ARG ) start_POSTSUPERSCRIPT italic_i + 1 end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( 2 italic_w ) start_POSTSUPERSCRIPT italic_j - italic_i end_POSTSUPERSCRIPT .
On the other hand, I I 𝐼 𝐼 II italic_I italic_I is estimated as:
I I := ∫ 2 w 2 F λ ( u + w ) f λ ( u ) j d u ≤ ∫ 2 w 2 F λ ( u ) f λ ( u ) j d u = 1 j + 1 f λ ( 2 w ) j + 1 , assign 𝐼 𝐼 superscript subscript 2 𝑤 2 subscript 𝐹 𝜆 𝑢 𝑤 subscript 𝑓 𝜆 superscript 𝑢 𝑗 differential-d 𝑢 subscript superscript 2 2 𝑤 subscript 𝐹 𝜆 𝑢 subscript 𝑓 𝜆 superscript 𝑢 𝑗 differential-d 𝑢 1 𝑗 1 subscript 𝑓 𝜆 superscript 2 𝑤 𝑗 1 \displaystyle II:=\int_{2w}^{2}F_{\lambda}(u+w)f_{\lambda}(u)^{j}\,\mathrm{d}u%\leq\int^{2}_{2w}F_{\lambda}(u)f_{\lambda}(u)^{j}\mathrm{d}u=\frac{1}{j+1}f_{%\lambda}(2w)^{j+1}, italic_I italic_I := ∫ start_POSTSUBSCRIPT 2 italic_w end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u + italic_w ) italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT roman_d italic_u ≤ ∫ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 italic_w end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT roman_d italic_u = divide start_ARG 1 end_ARG start_ARG italic_j + 1 end_ARG italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( 2 italic_w ) start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT ,
where we used the monotonicity of F 𝐹 F italic_F and the fact that f ′ = − F superscript 𝑓 ′ 𝐹 f^{\prime}=-F italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = - italic_F . This completes the proof.∎
Lemma 3.5. Fix m ≥ 2 𝑚 2 m\geq 2 italic_m ≥ 2 . For all 1 ≤ k ≤ m − 1 1 𝑘 𝑚 1 1\leq k\leq m-1 1 ≤ italic_k ≤ italic_m - 1 and ∑ i = 1 m − k u i ≤ 2 subscript superscript 𝑚 𝑘 𝑖 1 subscript 𝑢 𝑖 2 \sum^{m-k}_{i=1}u_{i}\leq 2 ∑ start_POSTSUPERSCRIPT italic_m - italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 2 with 0 ≤ u i ≤ 2 0 subscript 𝑢 𝑖 2 0\leq u_{i}\leq 2 0 ≤ italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 2 :
∫⋯∫ ∑ i = m − k + 1 m u i ≤ 2 ∏ r = m − k + 1 m F λ ( u r + u r − 1 2 ) d u r ≤ ∑ i = 0 k c i k ( k − i ) ! ( 4 log λ ) i f λ ( u m − k ) k − i subscript multiple-integral subscript superscript 𝑚 𝑖 𝑚 𝑘 1 subscript 𝑢 𝑖 2 subscript superscript product 𝑚 𝑟 𝑚 𝑘 1 subscript 𝐹 𝜆 subscript 𝑢 𝑟 subscript 𝑢 𝑟 1 2 d subscript 𝑢 𝑟 subscript superscript 𝑘 𝑖 0 subscript superscript 𝑐 𝑘 𝑖 𝑘 𝑖 superscript 4 𝜆 𝑖 subscript 𝑓 𝜆 superscript subscript 𝑢 𝑚 𝑘 𝑘 𝑖 \idotsint\limits_{\sum^{m}_{i=m-k+1}u_{i}\leq 2}\,\,\prod^{m}_{r=m-k+1}F_{%\lambda}\big{(}u_{r}+\frac{u_{r-1}}{2}\big{)}\,\mathrm{d}u_{r}\leq\sum^{k}_{i=%0}\frac{c^{k}_{i}}{(k-i)!}\left(\frac{4}{\log\lambda}\right)^{i}f_{\lambda}%\left(u_{m-k}\right)^{k-i} ∫⋯∫ start_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = italic_m - italic_k + 1 end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 2 end_POSTSUBSCRIPT ∏ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r = italic_m - italic_k + 1 end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + divide start_ARG italic_u start_POSTSUBSCRIPT italic_r - 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) roman_d italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ≤ ∑ start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT divide start_ARG italic_c start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ( italic_k - italic_i ) ! end_ARG ( divide start_ARG 4 end_ARG start_ARG roman_log italic_λ end_ARG ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_m - italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_k - italic_i end_POSTSUPERSCRIPT (3.15)
where c i k subscript superscript 𝑐 𝑘 𝑖 c^{k}_{i} italic_c start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are combinatorial coefficients defined inductively by
c 0 0 subscript superscript 𝑐 0 0 \displaystyle c^{0}_{0} italic_c start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 ; c i k = 0 for i > k and c i k + 1 = ∑ j = 0 i c j k for i ≤ k + 1 . formulae-sequence absent 1 formulae-sequence subscript superscript 𝑐 𝑘 𝑖 0 for i > k and
subscript superscript 𝑐 𝑘 1 𝑖 subscript superscript 𝑖 𝑗 0 subscript superscript 𝑐 𝑘 𝑗 for i ≤ k + 1
\displaystyle=1;\,c^{k}_{i}=0\,\text{ for $i>k$}\qquad\text{and}\qquad c^{k+1}%_{i}=\sum^{i}_{j=0}c^{k}_{j}\qquad\text{ for $i\leq k+1$}. = 1 ; italic_c start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 for italic_i > italic_k and italic_c start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∑ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT for italic_i ≤ italic_k + 1 . (3.16)
Proof. The proof here is an adaptation of the induction scheme of Lemma 3.9 in [ CZ23 ] .When k = 1 𝑘 1 k=1 italic_k = 1 , the statement follows from Lemma 3.4 for j = 0 𝑗 0 j=0 italic_j = 0 and w = u r − 1 2 𝑤 subscript 𝑢 𝑟 1 2 w=\frac{u_{r-1}}{2} italic_w = divide start_ARG italic_u start_POSTSUBSCRIPT italic_r - 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG .Assume the statement holds for some k 𝑘 k italic_k such that 1 ≤ k ≤ m − 2 1 𝑘