On the moments of the mass of shrinking balls under the Critical 2⁢𝑑 Stochastic Heat Flow (2024)

\MakePerPage

[2]footnote

Ziyang LiuDepartment of Mathematics
University of Warwick
Coventry CV4 7AL, UK
Ziyang.Liu.1@warwick.ac.uk
andNikos ZygourasDepartment of Mathematics
University of Warwick
Coventry CV4 7AL, UK
N.Zygouras@warwick.ac.uk

(Date: October 18, 2024)

Abstract.

The Critical 2d2𝑑2d2 italic_d Stochastic Heat Flow (SHF) is a measure valued stochastic process on 2superscript2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT thatdefines a non-trivial solution to the two-dimensional stochastic heat equation with multiplicativespace-time noise. Its one-time marginals are a.s. singular with respect to the Lebesgue measure,meaning that the mass they assign to shrinking balls decays to zero faster than their Lebesgue volume.In this work we explore the intermittency properties of the Critical 2d SHF by studying the asymptotics of the hhitalic_h-th momentof the mass that it assigns to shrinking balls of radius ε𝜀\varepsilonitalic_ε and we determine that its ratio to the Lebesgue volumeis of order (log1ε)(h2)superscript1𝜀binomial2(\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 up to possible lower order corrections.

1. Introduction

The Critical 2d2𝑑2d2 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)

tu=12Δu+βξu,t>0,x2,formulae-sequencesubscript𝑡𝑢12Δ𝑢𝛽𝜉𝑢formulae-sequence𝑡0𝑥superscript2\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 ξ𝜉\xiitalic_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 ξε(t,x):=1ε22j(xyε)ξ(t,dy)assignsuperscript𝜉𝜀𝑡𝑥1superscript𝜀2subscriptsuperscript2𝑗𝑥𝑦𝜀𝜉𝑡d𝑦\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 uεsuperscript𝑢𝜀u^{\varepsilon}italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT, which in fact can also be represented by a Feynman-Kac formula as

uε(t,x)=𝐄x[exp(β0tξε(ts,Bs)dsβ2t2jεL2(2)2)],superscript𝑢𝜀𝑡𝑥subscript𝐄𝑥delimited-[]𝛽superscriptsubscript0𝑡subscript𝜉𝜀𝑡𝑠subscript𝐵𝑠differential-d𝑠superscript𝛽2𝑡2superscriptsubscriptnormsubscript𝑗𝜀superscript𝐿2superscript22\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 Bssubscript𝐵𝑠B_{s}italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT being a two-dimensional Brownian motion whose expectation when starting from x2𝑥superscript2x\in\mathbb{R}^{2}italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is denoted by 𝐄xsubscript𝐄𝑥\boldsymbol{\mathrm{E}}_{x}bold_E start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and jε(x):=1ε2j(xε)assignsubscript𝑗𝜀𝑥1superscript𝜀2𝑗𝑥𝜀j_{\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 ε0𝜀0\varepsilon\to 0italic_ε → 0. We will see that for this to be the case a precise choice of β𝛽\betaitalic_β depending on ε𝜀\varepsilonitalic_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:

ZM,Nβ(x,y):=E[exp(n=M+1N1(βω(n,Sn)λ(β)))𝟙{SN=y}|SM=x],assignsuperscriptsubscript𝑍𝑀𝑁𝛽𝑥𝑦Edelimited-[]conditionalsubscriptsuperscript𝑁1𝑛𝑀1𝛽𝜔𝑛subscript𝑆𝑛𝜆𝛽subscript1subscript𝑆𝑁𝑦subscript𝑆𝑀𝑥\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 (Sn)n0subscriptsubscript𝑆𝑛𝑛0(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 PP\mathrm{P}roman_P and EE\mathrm{E}roman_E and (ωn,x)n,x2subscriptsubscript𝜔𝑛𝑥formulae-sequence𝑛𝑥superscript2(\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 00, variance 1111 and finite log-moment generating function λ(β):=log𝔼[eβω]<assign𝜆𝛽𝔼delimited-[]superscript𝑒𝛽𝜔\lambda(\beta):=\log{\mathbb{E}}\big{[}e^{\beta\omega}\big{]}<\inftyitalic_λ ( italic_β ) := roman_log blackboard_E [ italic_e start_POSTSUPERSCRIPT italic_β italic_ω end_POSTSUPERSCRIPT ] < ∞, for β𝛽\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 β𝛽\betaitalic_β, which modulates the strength of the noise. In the DPRE regularisation theCritical 2d SHF emerges through the choice of β=βN𝛽subscript𝛽𝑁\beta=\beta_{N}italic_β = italic_β start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT such that

σN2:=eλ(2βN)2λ(βN)1=πlogN(1+ϑ+o(1)logN),assignsuperscriptsubscript𝜎𝑁2superscript𝑒𝜆2subscript𝛽𝑁2𝜆subscript𝛽𝑁1𝜋𝑁1italic-ϑ𝑜1𝑁\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)𝑜1o(1)italic_o ( 1 ) denotes asymptotically negligible corrections as N𝑁N\to\inftyitalic_N → ∞. In the continuous approximation,β:=βεassign𝛽subscript𝛽𝜀\beta:=\beta_{\varepsilon}italic_β := italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ischosen as

βε2=2πlog1ε(1+ϱ+o(1)log1ε),subscriptsuperscript𝛽2𝜀2𝜋1𝜀1italic-ϱ𝑜11𝜀\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 ϱitalic-ϱ\varrhoitalic_ϱ is given as a function of the above ϑitalic-ϑ\varthetaitalic_ϑ and also depends on the mollifier j𝑗jitalic_j in a particular way. We refer toequation (1.38) in [CSZ19b] for the precise relation.

The Critical 2d2𝑑2d2 italic_d SHF was constructed in [CSZ23a] as the unique limit of the fields

𝒵N;s,tβ(dx,dy):=N4Z[Ns],[Nt]βN(Nx,Ny)dxdy,0s<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 2superscript2{\mathbb{R}}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT points totheir nearest even integer point on even2:={(z1,z2)2:z1+z22}assignsubscriptsuperscript2evenconditional-setsubscript𝑧1subscript𝑧2superscript2subscript𝑧1subscript𝑧22{\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 dxdyd𝑥d𝑦\mathrm{d}x\mathrm{d}yroman_d italic_x roman_d italic_y is the Lebesgue measure on 2×2superscript2superscript2{\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 βNsubscript𝛽𝑁\beta_{N}italic_β start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT be as in (1.5) for some fixed ϑitalic-ϑ\vartheta\in{\mathbb{R}}italic_ϑ ∈ blackboard_R and (𝒵N;s,tβ(dx,dy))0s<t<subscriptsuperscriptsubscript𝒵𝑁𝑠𝑡𝛽d𝑥d𝑦0𝑠𝑡\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_POSTSUBSCRIPTbe defined as in (1.6). Then as N𝑁N\rightarrow\inftyitalic_N → ∞, the process of random measures(𝒵N;s,tβ(dx,dy))0st<subscriptsuperscriptsubscript𝒵𝑁𝑠𝑡𝛽d𝑥d𝑦0𝑠𝑡({\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

𝒵ϑ=(𝒵s,tϑ(dx,dy))0st<,superscript𝒵italic-ϑsubscriptsuperscriptsubscript𝒵𝑠𝑡italic-ϑd𝑥d𝑦0𝑠𝑡\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.

𝒵ϑsuperscript𝒵italic-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

𝒵tϑ(𝟙,dy):=x2𝒵0,tϑ(dx,dy)=d𝒵tϑ(dx,𝟙):=y2𝒵0,tϑ(dx,dy),assignsuperscriptsubscript𝒵𝑡italic-ϑ1d𝑦subscript𝑥superscript2superscriptsubscript𝒵0𝑡italic-ϑd𝑥d𝑦superscript𝑑superscriptsubscript𝒵𝑡italic-ϑd𝑥1assignsubscript𝑦superscript2superscriptsubscript𝒵0𝑡italic-ϑd𝑥d𝑦\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):={y2:|yx|<ε},assign𝐵𝑥𝜀conditional-set𝑦superscript2𝑦𝑥𝜀B(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

𝒵tϑ(B(x,ε)):=yB(x,ε)𝒵tϑ(𝟙,dy),assignsuperscriptsubscript𝒵𝑡italic-ϑ𝐵𝑥𝜀subscript𝑦𝐵𝑥𝜀superscriptsubscript𝒵𝑡italic-ϑ1d𝑦\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>0𝑡0t>0italic_t > 0 and ϑitalic-ϑ\vartheta\in\mathbb{R}italic_ϑ ∈ blackboard_R,

-a.s.limε0𝒵tϑ(B(x,ε))Vol(B(x,ε))=limε01πε2𝒵tϑ(B(x,ε))=0for Lebesgue a.e.x2.formulae-sequence-a.s.subscript𝜀0superscriptsubscript𝒵𝑡italic-ϑ𝐵𝑥𝜀Vol𝐵𝑥𝜀subscript𝜀01𝜋superscript𝜀2superscriptsubscript𝒵𝑡italic-ϑ𝐵𝑥𝜀0for Lebesgue a.e.x2\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 ε0𝜀0\varepsilon\to 0italic_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

𝒵tϑ(φ):=2φ(x)𝒵tϑ(dx,𝟙),assignsubscriptsuperscript𝒵italic-ϑ𝑡𝜑subscriptsuperscript2𝜑𝑥superscriptsubscript𝒵𝑡italic-ϑd𝑥1\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 ϕitalic-ϕ\phiitalic_ϕ on 2superscript2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.Our result then is the following:

Theorem 1.2.

Let 𝒰B(0,ε)()subscript𝒰𝐵0𝜀{\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 2superscript2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT:

𝒰B(0,ε)():=1πε2 1B(0,ε)()whereB(0,ε):={y2:|y|<ε},formulae-sequenceassignsubscript𝒰𝐵0𝜀1𝜋superscript𝜀2subscript1𝐵0𝜀assignwhere𝐵0𝜀conditional-set𝑦superscript2𝑦𝜀{\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 𝒵tϑ(𝒰B(0,ε))superscriptsubscript𝒵𝑡italic-ϑsubscript𝒰𝐵0𝜀\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()=𝒰B(0,ε)()𝜑subscript𝒰𝐵0𝜀\varphi(\cdot)={\mathcal{U}}_{B(0,\varepsilon)}(\cdot)italic_φ ( ⋅ ) = caligraphic_U start_POSTSUBSCRIPT italic_B ( 0 , italic_ε ) end_POSTSUBSCRIPT ( ⋅ ).For all h22h\geq 2italic_h ≥ 2, t>0𝑡0t>0italic_t > 0 and ϑitalic-ϑ\vartheta\in\mathbb{R}italic_ϑ ∈ blackboard_R there exist a constant C=C(h,ϑ,t)𝐶𝐶italic-ϑ𝑡C=C(h,\vartheta,t)italic_C = italic_C ( italic_h , italic_ϑ , italic_t ) such that

C(log1ε)(h2)𝔼[(𝒵tϑ(𝒰B(0,ε)))h](log1ε)(h2)+o(1),𝐶superscript1𝜀binomial2𝔼delimited-[]superscriptsuperscriptsubscript𝒵𝑡italic-ϑsubscript𝒰𝐵0𝜀superscript1𝜀binomial2𝑜1\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)𝑜1o(1)italic_o ( 1 ) representing terms that go to 00 as ε0𝜀0\varepsilon\to 0italic_ε → 0.

We note that for h=22h=2italic_h = 2 the correlation structure of the Critical 2d SHF already providesthe sharp asymptotic

𝔼[(𝒵tϑ(gε2))2]Ctlog1ε,asε0,similar-to𝔼delimited-[]superscriptsuperscriptsubscript𝒵𝑡italic-ϑsubscript𝑔superscript𝜀22subscript𝐶𝑡1𝜀asε0\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 HamiltonianΔ+1i<jhδ0(xixj)Δsubscript1𝑖𝑗subscript𝛿0subscript𝑥𝑖subscript𝑥𝑗-\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 (2)hsuperscriptsuperscript2(\mathbb{R}^{2})^{h}( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPTknown as the delta-Bose gas [AFH+92, DFT94, DR04]; δ0()subscript𝛿0\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 operatorsΔ+1i<jhβε2δε(xixj)Δsubscript1𝑖𝑗superscriptsubscript𝛽𝜀2subscript𝛿𝜀subscript𝑥𝑖subscript𝑥𝑗-\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 (2)hsuperscriptsuperscript2(\mathbb{R}^{2})^{h}( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT,where βε2superscriptsubscript𝛽𝜀2\beta_{\varepsilon}^{2}italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is as in (1.5) and δεsubscript𝛿𝜀\delta_{\varepsilon}italic_δ start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT a mollification of the delta function with a jεsubscript𝑗𝜀j_{\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 βε2superscriptsubscript𝛽𝜀2\beta_{\varepsilon}^{2}italic_β start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPTin (1.5) being equal to 2π2𝜋2\pi2 italic_π.It is well known that independent Brownian motions in dimension 2222 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=22h=2italic_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 hhitalic_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 situationin the case when the starting points of the Brownian motion are spread out rather than beingconcentrated in a ε𝜀\varepsilonitalic_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 ballsε0𝜀0\varepsilon\to 0italic_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 ε0𝜀0\varepsilon\to 0italic_ε → 0 proportionally to(log1ε)(h2)superscript1𝜀binomial2(\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(log1ε)(h2)𝔼[(𝒵tϑ(𝒰B(0,ε)))h]1superscript1𝜀binomial2𝔼delimited-[]superscriptsuperscriptsubscript𝒵𝑡italic-ϑsubscript𝒰𝐵0𝜀\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\inftydivide 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|logε|1|logloglogε|superscript𝜀1𝜀|\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 μ𝜇\muitalic_μ on dsuperscript𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is determined via the exponent ξ(h)𝜉\xi(h)italic_ξ ( italic_h ) inthe moment asymptotics

𝔼[μ(B(x,ε))h]εξ(h),asε0,similar-to𝔼delimited-[]𝜇superscript𝐵𝑥𝜀superscript𝜀𝜉asε0\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]01h\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 μ𝜇\muitalic_μ is said to exhibit multifractality if the exponent ξ(h)𝜉\xi(h)italic_ξ ( italic_h ) is a nonlinearfunction of hhitalic_h. Our result that

𝔼[𝒵tϑ(B(x,ε))h]ε2h(log1ε)h(h1)2+o(1),asε0for2h,similar-to𝔼delimited-[]superscriptsubscript𝒵𝑡italic-ϑsuperscript𝐵𝑥𝜀superscript𝜀2superscript1𝜀12𝑜1asε0for2h\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 𝒞0superscript𝒞limit-from0{\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]01h\in[0,1]italic_h ∈ [ 0 , 1 ].We conjecture that asymptotic (1.15) extends to h[0,1]01h\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 SHF

In 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[𝒵s,tϑ(dx,dy)]=12g12(ts)(yx)dxdy,𝔼delimited-[]subscriptsuperscript𝒵italic-ϑ𝑠𝑡d𝑥d𝑦12subscript𝑔12𝑡𝑠𝑦𝑥d𝑥d𝑦{\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 gt(x)=12πte|x|22tsubscript𝑔𝑡𝑥12𝜋𝑡superscript𝑒superscript𝑥22𝑡g_{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

ov[𝒵s,tϑ(dx,dy),𝒵s,tϑ(dx,dy)]ovsubscriptsuperscript𝒵italic-ϑ𝑠𝑡d𝑥d𝑦subscriptsuperscript𝒵italic-ϑ𝑠𝑡dsuperscript𝑥dsuperscript𝑦\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 ) ]=12Ktsϑ(x,x;y,y)dxdydxdy,absent12superscriptsubscript𝐾𝑡𝑠italic-ϑ𝑥superscript𝑥𝑦superscript𝑦d𝑥d𝑦dsuperscript𝑥dsuperscript𝑦\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

Ktϑ(x,x;y,y):=πgt4(y+y2x+x2)0<a<b<tga(xx)Gϑ(ba)gtb(yy)dadb.assignsuperscriptsubscript𝐾𝑡italic-ϑ𝑥superscript𝑥𝑦superscript𝑦𝜋subscript𝑔𝑡4𝑦superscript𝑦2𝑥superscript𝑥2subscriptdouble-integral0𝑎𝑏𝑡subscript𝑔𝑎superscript𝑥𝑥subscript𝐺italic-ϑ𝑏𝑎subscript𝑔𝑡𝑏superscript𝑦𝑦differential-d𝑎differential-d𝑏\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 Gϑ(t)subscript𝐺italic-ϑ𝑡G_{\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 Gϑ(t)subscript𝐺italic-ϑ𝑡G_{\vartheta}(t)italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) is

Gϑ(t)=0e(ϑγ)ssts1Γ(s+1)ds,subscript𝐺italic-ϑ𝑡superscriptsubscript0superscript𝑒italic-ϑ𝛾𝑠𝑠superscript𝑡𝑠1Γ𝑠1differential-d𝑠\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:=0logueudu0.577assign𝛾superscriptsubscript0𝑢superscript𝑒𝑢d𝑢0.577\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)Γ𝑠\Gamma(s)roman_Γ ( italic_s ) is the Gamma function.For t(0,1)𝑡01t\in(0,1)italic_t ∈ ( 0 , 1 ) (2.4) can be written as

Gϑ(t)=0eϑsfs(t)ds,subscript𝐺italic-ϑ𝑡superscriptsubscript0superscript𝑒italic-ϑ𝑠subscript𝑓𝑠𝑡differential-d𝑠\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 fs(t)subscript𝑓𝑠𝑡f_{s}(t)italic_f start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ) is the density of the Dickman subordinator (Ys)s>0subscriptsubscript𝑌𝑠𝑠0(Y_{s})_{s>0}( italic_Y start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_s > 0 end_POSTSUBSCRIPT – a jump process with Lévy measurex1𝟙x(0,1)dxsuperscript𝑥1subscript1𝑥01d𝑥x^{-1}\mathds{1}_{x\in(0,1)}\mathrm{d}xitalic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_x ∈ ( 0 , 1 ) end_POSTSUBSCRIPT roman_d italic_x – and is given by

fs(t)={sts1eγsΓ(s+1),fort(0,1],sts1eγsΓ(s+1)sts101fs(a)(1+a)sda,fort[1,).subscript𝑓𝑠𝑡cases𝑠superscript𝑡𝑠1superscript𝑒𝛾𝑠Γ𝑠1fort(0,1]otherwiseotherwise𝑠superscript𝑡𝑠1superscript𝑒𝛾𝑠Γ𝑠1𝑠superscript𝑡𝑠1superscriptsubscript01subscript𝑓𝑠𝑎superscript1𝑎𝑠differential-d𝑎fort[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 Gϑ(t)subscript𝐺italic-ϑ𝑡G_{\vartheta}(t)italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) for t1𝑡1t\leq 1italic_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 Gϑ(t)subscript𝐺italic-ϑ𝑡G_{\vartheta}(t)italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) be as in (2.4) for t>0𝑡0t>0italic_t > 0. Then for λ>eϑγ𝜆superscript𝑒italic-ϑ𝛾\lambda>e^{\vartheta-\gamma}italic_λ > italic_e start_POSTSUPERSCRIPT italic_ϑ - italic_γ end_POSTSUPERSCRIPT we have that

0eλtGϑ(t)dt=1logλϑ+γ.superscriptsubscript0superscript𝑒𝜆𝑡subscript𝐺italic-ϑ𝑡differential-d𝑡1𝜆italic-ϑ𝛾\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:

    0Gϑ(t)eλtdtsuperscriptsubscript0subscript𝐺italic-ϑ𝑡superscript𝑒𝜆𝑡differential-d𝑡\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=00e(ϑγ)sts1Γ(s)eλtdsdtabsentsuperscriptsubscript0superscriptsubscript0superscript𝑒italic-ϑ𝛾𝑠superscript𝑡𝑠1Γ𝑠superscript𝑒𝜆𝑡differential-d𝑠differential-d𝑡\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(0ts1eλtdt)e(ϑγ)sΓ(s)dsabsentsuperscriptsubscript0superscriptsubscript0superscript𝑡𝑠1superscript𝑒𝜆𝑡differential-d𝑡superscript𝑒italic-ϑ𝛾𝑠Γ𝑠differential-d𝑠\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(1λs0ts1etdt)e(ϑγ)sΓ(s)dsabsentsuperscriptsubscript01superscript𝜆𝑠superscriptsubscript0superscript𝑡𝑠1superscript𝑒𝑡differential-d𝑡superscript𝑒italic-ϑ𝛾𝑠Γ𝑠differential-d𝑠\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
    =01λse(ϑγ)sds=0e(logλϑ+γ)sdsabsentsubscriptsuperscript01superscript𝜆𝑠superscript𝑒italic-ϑ𝛾𝑠differential-d𝑠subscriptsuperscript0superscript𝑒𝜆italic-ϑ𝛾𝑠differential-d𝑠\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
    =1logλϑ+γ.absent1𝜆italic-ϑ𝛾\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 Gϑsubscript𝐺italic-ϑG_{\vartheta}italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT, which were established in [CSZ19a]

Proposition 2.2.

For any ϑitalic-ϑ\vartheta\in{\mathbb{R}}italic_ϑ ∈ blackboard_R, the function Gϑ(t)subscript𝐺italic-ϑ𝑡G_{\vartheta}(t)italic_G start_POSTSUBSCRIPT italic_ϑ end_POSTSUBSCRIPT ( italic_t ) is continuous and strictly positive for t(0,1]𝑡01t\in(0,1]italic_t ∈ ( 0 , 1 ]. As t0𝑡0t\downarrow 0italic_t ↓ 0 we have the asymptotic,

Gϑ(t)=1t(log1t)2{1+2ϑlog1t+O(1(log1t)2)}.subscript𝐺italic-ϑ𝑡1𝑡superscript1𝑡212italic-ϑ1𝑡𝑂1superscript1𝑡2G_{\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 hhitalic_h-momentand demystify it afterwards. The formula is:

𝔼[(𝒵tϑ(φ))h]=m0(2π)m{{i1,j1},,{im,jm}{1,,h}2with{ik,jk}{ik+1,jk+1}fork=1,,m1(2)hd𝒙ϕh(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 )
0a1<b1<<am<bmtx1,y1,,xm,ym2ga12(x1xi1)ga12(x1xj1)r=1mGϑ(brar)gbrar4(yrxr)  1𝒮ir,jrsubscriptdouble-integral0subscript𝑎1subscript𝑏1subscript𝑎𝑚subscript𝑏𝑚𝑡subscript𝑥1subscript𝑦1subscript𝑥𝑚subscript𝑦𝑚superscript2subscript𝑔subscript𝑎12subscript𝑥1superscript𝑥subscript𝑖1subscript𝑔subscript𝑎12subscript𝑥1superscript𝑥subscript𝑗1superscriptsubscriptproduct𝑟1𝑚subscript𝐺italic-ϑsubscript𝑏𝑟subscript𝑎𝑟subscript𝑔subscript𝑏𝑟subscript𝑎𝑟4subscript𝑦𝑟subscript𝑥𝑟subscript1subscript𝒮subscript𝑖𝑟subscript𝑗𝑟\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
×(1rm1gar+1b𝗉(ir+1)2(xr+1y𝗉(ir+1))gar+1b𝗉(jr+1)2(xr+1y𝗉(jr+1)))d𝒙d𝒚dadbabsentsubscriptproduct1𝑟𝑚1subscript𝑔subscript𝑎𝑟1subscript𝑏𝗉subscript𝑖𝑟12subscript𝑥𝑟1subscript𝑦𝗉subscript𝑖𝑟1subscript𝑔subscript𝑎𝑟1subscript𝑏𝗉subscript𝑗𝑟12subscript𝑥𝑟1subscript𝑦𝗉subscript𝑗𝑟1d𝒙d𝒚d𝑎d𝑏\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 ϕh(𝒙):=ϕ(x1)ϕ(xh)assignsuperscriptitalic-ϕtensor-productabsent𝒙italic-ϕsuperscript𝑥1italic-ϕsuperscript𝑥\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 ),𝒮ir,jrsubscript𝒮subscript𝑖𝑟subscript𝑗𝑟{\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 irsubscript𝑖𝑟i_{r}italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and jrsubscript𝑗𝑟j_{r}italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT, only, are involved in collisions in the time interval(ar,br)subscript𝑎𝑟subscript𝑏𝑟(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 xrsubscript𝑥𝑟x_{r}italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and ending at positions yrsubscript𝑦𝑟y_{r}italic_y start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and for a pair {ir,jr}subscript𝑖𝑟subscript𝑗𝑟\{i_{r},j_{r}\}{ italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT } we define

𝗉(ir)𝗉subscript𝑖𝑟\displaystyle\mathsf{p}(i_{r})sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ):=i(r)with(r):=max{0<r:𝟙𝒮i,j=1andir{i,j}}formulae-sequenceassignabsentsubscript𝑖𝑟withassign𝑟:0𝑟subscript1subscript𝒮subscript𝑖subscript𝑗1andsubscript𝑖𝑟subscript𝑖subscript𝑗\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(jr)𝗉subscript𝑗𝑟\mathsf{p}(j_{r})sansserif_p ( italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ).In other words, 𝗉(ir)𝗉subscript𝑖𝑟\mathsf{p}(i_{r})sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) is the last time before r𝑟ritalic_r that Brownian motion B(ir)superscript𝐵subscript𝑖𝑟B^{(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(ir)=0𝗉subscript𝑖𝑟0\mathsf{p}(i_{r})=0sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = 0 then (b𝗉(ir),y𝗉(ir)):=(0,xir)assignsubscript𝑏𝗉subscript𝑖𝑟subscript𝑦𝗉subscript𝑖𝑟0superscript𝑥subscript𝑖𝑟(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[(2ϕ(x)uε(t,x)dx)h]=(2)hϕh(𝒙)𝐄𝒙h[(βε21i<jh0tJε(Bs(i)Bs(j))ds)]d𝒙𝔼delimited-[]superscriptsubscriptsuperscript2italic-ϕ𝑥superscript𝑢𝜀𝑡𝑥differential-d𝑥subscriptsuperscriptsuperscript2superscriptitalic-ϕtensor-productabsent𝒙superscriptsubscript𝐄𝒙tensor-productabsentdelimited-[]superscriptsubscript𝛽𝜀2subscript1𝑖𝑗superscriptsubscript0𝑡subscript𝐽𝜀superscriptsubscript𝐵𝑠𝑖superscriptsubscript𝐵𝑠𝑗differential-d𝑠differential-d𝒙\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=(x1,,xh)𝒙superscript𝑥1superscript𝑥\boldsymbol{x}=(x^{1},...,x^{h})bold_italic_x = ( italic_x start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_x start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ). Jε(x):=βε21ε2J(xε)assignsubscript𝐽𝜀𝑥superscriptsubscript𝛽𝜀21superscript𝜀2𝐽𝑥𝜀J_{\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=jj𝐽𝑗𝑗J=j*jitalic_J = italic_j ∗ italic_j and j𝑗jitalic_j as in (1.2),approximates a delta function when ε0𝜀0\varepsilon\to 0italic_ε → 0. When βεsubscript𝛽𝜀\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 ε0𝜀0\varepsilon\to 0italic_ε → 0comes from configurations where the Brownian motions B(1),,B(h)superscript𝐵1superscript𝐵B^{(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 ϕitalic-ϕ\phiitalic_ϕ is 𝒰B(0,ε)():=1πε2 1B(0,ε)()assignsubscript𝒰𝐵0𝜀1𝜋superscript𝜀2subscript1𝐵0𝜀{\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 ϕitalic-ϕ\phiitalic_ϕ being a heat kernel approximation of the delta functionand look into the asymptotics of

𝒵εϑ,h:=𝔼[(𝒵1ϑ(gε22))h]withgε22(x)=1πε2e|x|2ε2,formulae-sequenceassignsuperscriptsubscript𝒵𝜀italic-ϑ𝔼delimited-[]superscriptsuperscriptsubscript𝒵1italic-ϑsubscript𝑔superscript𝜀22withsubscript𝑔superscript𝜀22𝑥1𝜋superscript𝜀2superscript𝑒superscript𝑥2superscript𝜀2\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[(𝒵1ϑ(𝒰B(0,ε)))h]𝔼delimited-[]superscriptsuperscriptsubscript𝒵1italic-ϑsubscript𝒰𝐵0𝜀{\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=1𝑡1t=1italic_t = 1.

Let us write the series expression for 𝒵εϑ,hsuperscriptsubscript𝒵𝜀italic-ϑ\mathscr{Z}_{\varepsilon}^{\vartheta,h}script_Z start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT.For every i𝑖iitalic_i, we integrate,gε22(xi)subscript𝑔superscript𝜀22superscript𝑥𝑖g_{\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 xisuperscript𝑥𝑖x^{i}italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT (see Figure 1):

2gε22(xi)ga𝗋(i)2(x𝗋(i)xi)dxi=ga𝗋(i)+ε22(y𝗋(i)xi),subscriptsuperscript2subscript𝑔superscript𝜀22superscript𝑥𝑖subscript𝑔subscript𝑎𝗋𝑖2subscript𝑥𝗋𝑖superscript𝑥𝑖differential-dsuperscript𝑥𝑖subscript𝑔subscript𝑎𝗋𝑖superscript𝜀22subscript𝑦𝗋𝑖superscript𝑥𝑖\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)𝗋𝑖\mathsf{r}(i)sansserif_r ( italic_i ) the index which determines the point (ar,xr),r=1,,mformulae-sequencesubscript𝑎𝑟subscript𝑥𝑟𝑟1𝑚(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_mthat is connected to (0,xi)0superscript𝑥𝑖(0,x^{i})( 0 , italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ). Performing all such integrations over the initial points xi,i=1,,hformulae-sequencesuperscript𝑥𝑖𝑖1x^{i},i=1,...,hitalic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_i = 1 , … , italic_hand shifting the time variables a1,b1,,ar,brsubscript𝑎1subscript𝑏1subscript𝑎𝑟subscript𝑏𝑟a_{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 ε2superscript𝜀2\varepsilon^{2}italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, we arrive at the following formula,which is depicted in Figure 2:

𝒵εϑ,h=m0(2π)m{{i1,j1},,{im,jm}{1,,h}2with{ik,jk}{ik+1,jk+1}fork=1,,m1\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
ε2a1<b1<<am<bm1+ε2x1,y1,,xm,ym2ga12(x1)2r=1mGϑ(brar)gbrar4(yrxr)  1𝒮ir,jrsubscriptdouble-integralsuperscript𝜀2subscript𝑎1subscript𝑏1subscript𝑎𝑚subscript𝑏𝑚1superscript𝜀2subscript𝑥1subscript𝑦1subscript𝑥𝑚subscript𝑦𝑚superscript2subscript𝑔subscript𝑎12superscriptsubscript𝑥12superscriptsubscriptproduct𝑟1𝑚subscript𝐺italic-ϑsubscript𝑏𝑟subscript𝑎𝑟subscript𝑔subscript𝑏𝑟subscript𝑎𝑟4subscript𝑦𝑟subscript𝑥𝑟subscript1subscript𝒮subscript𝑖𝑟subscript𝑗𝑟\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)
×(1rm1gar+1b𝗉(ir+1)2(xr+1y𝗉(ir+1))gar+1b𝗉(jr+1)2(xr+1y𝗉(jr+1)))d𝒙d𝒚dadb,absentsubscriptproduct1𝑟𝑚1subscript𝑔subscript𝑎𝑟1subscript𝑏𝗉subscript𝑖𝑟12subscript𝑥𝑟1subscript𝑦𝗉subscript𝑖𝑟1subscript𝑔subscript𝑎𝑟1subscript𝑏𝗉subscript𝑗𝑟12subscript𝑥𝑟1subscript𝑦𝗉subscript𝑗𝑟1d𝒙d𝒚d𝑎d𝑏\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(ir)=0𝗉subscript𝑖𝑟0\mathsf{p}(i_{r})=0sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = 0, then (b𝗉(ir),y𝗉(ir))=(0,0)subscript𝑏𝗉subscript𝑖𝑟subscript𝑦𝗉subscript𝑖𝑟00(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 bound

In 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 𝒵εϑ,hsuperscriptsubscript𝒵𝜀italic-ϑ\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>0𝛿0\delta>0italic_δ > 0, h22h\geq 2italic_h ≥ 2 and ϑitalic-ϑ\vartheta\in{\mathbb{R}}italic_ϑ ∈ blackboard_R, then

𝒵εϑ,h(log1ε)(h2)+o(1),asε0.subscriptsuperscript𝒵italic-ϑ𝜀superscript1𝜀binomial2𝑜1asε0\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:

    𝒰B(0,ε)()=1πε2𝟙B(0,ε)()egε2/2().subscript𝒰𝐵0𝜀1𝜋superscript𝜀2subscript1𝐵0𝜀𝑒subscript𝑔superscript𝜀22\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[(𝒵1ϑ(𝒰B(0,ε)))h]eh𝒵εϑ,heh(log1ε)(h2)+o(1).𝔼delimited-[]superscriptsubscriptsuperscript𝒵italic-ϑ1subscript𝒰𝐵0𝜀superscript𝑒subscriptsuperscript𝒵italic-ϑ𝜀superscript𝑒superscript1𝜀binomial2𝑜1\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 ε0𝜀0\varepsilon\rightarrow 0italic_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 on𝒵εϑ,hsuperscriptsubscript𝒵𝜀italic-ϑ\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<10𝜀10<\varepsilon<10 < italic_ε < 1, the following estimate holds:

𝒵εϑ,hsubscriptsuperscript𝒵italic-ϑ𝜀\displaystyle\mathscr{Z}^{\vartheta,h}_{\varepsilon}script_Z start_POSTSUPERSCRIPT italic_ϑ , italic_h end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPTm0m,h,εabsentsubscript𝑚0subscript𝑚𝜀\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 0,h,ε=1subscript0𝜀1\mathscr{I}_{0,h,\varepsilon}=1script_I start_POSTSUBSCRIPT 0 , italic_h , italic_ε end_POSTSUBSCRIPT = 1, 1,h,ε=C(h2)log1εsubscript1𝜀𝐶binomial21𝜀\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>0𝐶0C>0italic_C > 0, and for m2𝑚2m\geq 2italic_m ≥ 2:

m,h,ε:=(h2)[(h2)1]m1∫⋯∫i(ui+vi)1+ε2,u1>ε21u11rm1Gϑ(vr)12(vr+ur)+ur+1Gϑ(vm)dudvassignsubscript𝑚𝜀binomial2superscriptdelimited-[]binomial21𝑚1subscriptmultiple-integralformulae-sequencesubscript𝑖subscript𝑢𝑖subscript𝑣𝑖1superscript𝜀2subscript𝑢1superscript𝜀21subscript𝑢1subscriptproduct1𝑟𝑚1subscript𝐺italic-ϑsubscript𝑣𝑟12subscript𝑣𝑟subscript𝑢𝑟subscript𝑢𝑟1subscript𝐺italic-ϑsubscript𝑣𝑚d𝑢d𝑣\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=0𝑚0m=0italic_m = 0 term in that formula is simply 1111. The m=1𝑚1m=1italic_m = 1 term is equal to:

    2πi,j{1,,h}x,y2ε2a<b1+ε2ga2(x)2Gϑ(ba)gba4(yx)dxdydadb.2𝜋subscript𝑖𝑗1subscriptdouble-integralsuperscript𝑥𝑦superscript2superscript𝜀2𝑎𝑏1superscript𝜀2subscript𝑔𝑎2superscript𝑥2subscript𝐺italic-ϑ𝑏𝑎subscript𝑔𝑏𝑎4𝑦𝑥d𝑥d𝑦d𝑎d𝑏\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 ε2a<b1+ε2()dadbsubscriptdouble-integralsuperscript𝜀2𝑎𝑏1superscript𝜀2differential-d𝑎differential-d𝑏\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 ε2a<b2()dadbsubscriptdouble-integralsuperscript𝜀2𝑎𝑏2differential-d𝑎differential-d𝑏\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 y𝑦yitalic_y, which gives

    2gba4(yx)dy=1.subscriptsuperscript2subscript𝑔𝑏𝑎4𝑦𝑥differential-d𝑦1\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 b𝑏bitalic_b:

    a2Gϑ(ba)dbC.superscriptsubscript𝑎2subscript𝐺italic-ϑ𝑏𝑎differential-d𝑏𝐶\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:

    Ci,j{1,,h}ε2a2,x2ga2(x)2dxda𝐶subscript𝑖𝑗1subscriptdouble-integralformulae-sequencesuperscript𝜀2𝑎2𝑥superscript2subscript𝑔𝑎2superscript𝑥2d𝑥d𝑎\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}aitalic_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(h2)ε2a2,x2ga2(x)2dxdaabsent𝐶binomial2subscriptdouble-integralformulae-sequencesuperscript𝜀2𝑎2𝑥superscript2subscript𝑔𝑎2superscript𝑥2differential-d𝑥differential-d𝑎\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(h2)ε22ga(0)daabsent𝐶binomial2superscriptsubscriptsuperscript𝜀22subscript𝑔𝑎0differential-d𝑎\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(h2)log(2ε2)C(h2)log(1ε).absent𝐶binomial22superscript𝜀2𝐶binomial21𝜀\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 m2𝑚2m\geq 2italic_m ≥ 2. We will follow the convention that b0=0subscript𝑏00b_{0}=0italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0. and recall the convention that if 𝗉(ir)=0𝗉subscript𝑖𝑟0\mathsf{p}(i_{r})=0sansserif_p ( italic_i start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = 0, then(b𝗉(ir),y𝗉(ir))=(0,0)subscript𝑏𝗉subscript𝑖𝑟subscript𝑦𝗉subscript𝑖𝑟00(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 ymsubscript𝑦𝑚y_{m}italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, which amounts to

    2gbmam4(ymxm)dym=1.subscriptsuperscript2subscript𝑔subscript𝑏𝑚subscript𝑎𝑚4subscript𝑦𝑚subscript𝑥𝑚differential-dsubscript𝑦𝑚1\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 xmsubscript𝑥𝑚x_{m}italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT.

    2gamb𝗆,𝗁,ε𝗉(𝗂𝗆)2(xmy𝗉(im))gamb𝗉(jm)2(xmy𝗉(jm))dxmsubscriptsuperscript2subscript𝑔subscript𝑎𝑚subscript𝑏subscript𝗆𝗁𝜀𝗉subscript𝗂𝗆2subscript𝑥𝑚subscript𝑦𝗉subscript𝑖𝑚subscript𝑔subscript𝑎𝑚subscript𝑏𝗉subscript𝑗𝑚2subscript𝑥𝑚subscript𝑦𝗉subscript𝑗𝑚differential-dsubscript𝑥𝑚\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
    =gam12(b𝗉(im)+b𝗉(jm))(y𝗉(im)y𝗉(jm))1/π2am(b𝗉(im)+b𝗉(jm))1/π(ambm1)+(ambm2),absentsubscript𝑔subscript𝑎𝑚12subscript𝑏𝗉subscript𝑖𝑚subscript𝑏𝗉subscript𝑗𝑚subscript𝑦𝗉subscript𝑖𝑚subscript𝑦𝗉subscript𝑗𝑚1𝜋2subscript𝑎𝑚subscript𝑏𝗉subscript𝑖𝑚subscript𝑏𝗉subscript𝑗𝑚1𝜋subscript𝑎𝑚subscript𝑏𝑚1subscript𝑎𝑚subscript𝑏𝑚2\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 b𝗉(im)subscript𝑏𝗉subscript𝑖𝑚b_{\mathsf{p}(i_{m})}italic_b start_POSTSUBSCRIPT sansserif_p ( italic_i start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT and b𝗉(jm)subscript𝑏𝗉subscript𝑗𝑚b_{\mathsf{p}(j_{m})}italic_b start_POSTSUBSCRIPT sansserif_p ( italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT may be before bm1subscript𝑏𝑚1b_{m-1}italic_b start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT and bm2subscript𝑏𝑚2b_{m-2}italic_b start_POSTSUBSCRIPT italic_m - 2 end_POSTSUBSCRIPT, respectively, but notafter and they cannot be both equal to just one of bm1subscript𝑏𝑚1b_{m-1}italic_b start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT or bm2subscript𝑏𝑚2b_{m-2}italic_b start_POSTSUBSCRIPT italic_m - 2 end_POSTSUBSCRIPT.

    The result then follows by iterating the same integration successively overym1,xm1,,y1,x1subscript𝑦𝑚1subscript𝑥𝑚1subscript𝑦1subscript𝑥1y_{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

    vi:=biaiandui:=aibi1.formulae-sequenceassignsubscript𝑣𝑖subscript𝑏𝑖subscript𝑎𝑖andassignsubscript𝑢𝑖subscript𝑎𝑖subscript𝑏𝑖1\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 (h2)[(h2)1]m1binomial2superscriptdelimited-[]binomial21𝑚1{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}𝑖𝑗\{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 v1,,vrsubscript𝑣1subscript𝑣𝑟v_{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>0𝐶0C>0italic_C > 0 such that for all λ>eϑγ𝜆superscript𝑒italic-ϑ𝛾\lambda>e^{\vartheta-\gamma}italic_λ > italic_e start_POSTSUPERSCRIPT italic_ϑ - italic_γ end_POSTSUPERSCRIPT, it holds

𝒵εϑ,hCe2λm=0m,h,ε(λ)subscriptsuperscript𝒵italic-ϑ𝜀𝐶superscript𝑒2𝜆superscriptsubscript𝑚0subscriptsuperscript𝜆𝑚𝜀\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 m,h,ε(λ):=m,h,εassignsubscriptsuperscript𝜆𝑚𝜀subscript𝑚𝜀\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=0,1𝑚01m=0,1italic_m = 0 , 1, and for m2𝑚2m\geq 2italic_m ≥ 2:

m,h,ε(λ):=(h2)[(h2)1]m1∫⋯∫iui2,u1>ε21u1r=2mFλ(ur+ur12)du,assignsubscriptsuperscript𝜆𝑚𝜀binomial2superscriptdelimited-[]binomial21𝑚1subscriptmultiple-integralformulae-sequencesubscript𝑖subscript𝑢𝑖2subscript𝑢1superscript𝜀21subscript𝑢1subscriptsuperscriptproduct𝑚𝑟2subscript𝐹𝜆subscript𝑢𝑟subscript𝑢𝑟12d𝑢\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

Fλ(w):=0eσwdσlog(λ+σ/2)ϑ+γ.assignsubscript𝐹𝜆𝑤subscriptsuperscript0superscript𝑒𝜎𝑤d𝜎𝜆𝜎2italic-ϑ𝛾F_{\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<2𝑚2m<2italic_m < 2 there is nothing to prove. For m2𝑚2m\geq 2italic_m ≥ 2, to simplify notationally, we extend the integral in (3.2) to i(ui+vi)<2subscript𝑖subscript𝑢𝑖subscript𝑣𝑖2\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>0𝜆0\lambda>0italic_λ > 0, which will be suitably chosen later onand we multiply (3.2) by e2λeλivi1superscript𝑒2𝜆superscript𝑒𝜆subscript𝑖subscript𝑣𝑖1e^{2\lambda}e^{-\lambda\sum_{i}v_{i}}\geq 1italic_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 ≥ 1to obtain the bound

    m,h,εe2λ(h2)[(h2)1]m1∫⋯∫i(ui+vi)2,u1>ε21u11rm1eλvrGϑ(vr)12(vr+ur)+ur+1Gϑ(vm)dudv.subscript𝑚𝜀superscript𝑒2𝜆binomial2superscriptdelimited-[]binomial21𝑚1subscriptmultiple-integralformulae-sequencesubscript𝑖subscript𝑢𝑖subscript𝑣𝑖2subscript𝑢1superscript𝜀21subscript𝑢1subscriptproduct1𝑟𝑚1superscript𝑒𝜆subscript𝑣𝑟subscript𝐺italic-ϑsubscript𝑣𝑟12subscript𝑣𝑟subscript𝑢𝑟subscript𝑢𝑟1subscript𝐺italic-ϑsubscript𝑣𝑚d𝑢d𝑣\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 v𝑣vitalic_v variables. Starting from vmsubscript𝑣𝑚v_{m}italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT we use the bound

    02Gϑ(vm)eλvmdvm02Gϑ(vm)dvmCsuperscriptsubscript02subscript𝐺italic-ϑsubscript𝑣𝑚superscript𝑒𝜆subscript𝑣𝑚differential-dsubscript𝑣𝑚superscriptsubscript02subscript𝐺italic-ϑsubscript𝑣𝑚differential-dsubscript𝑣𝑚𝐶\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 v𝑣vitalic_v-variables we usethat for any w>0𝑤0w>0italic_w > 0 we have the bound:

    02eλvGϑ(v)v/2+wdvsuperscriptsubscript02superscript𝑒𝜆𝑣subscript𝐺italic-ϑ𝑣𝑣2𝑤differential-d𝑣\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=020eσ(v/2+w)eλvGϑ(v)dσdvabsentsubscriptsuperscript20subscriptsuperscript0superscript𝑒𝜎𝑣2𝑤superscript𝑒𝜆𝑣subscript𝐺italic-ϑ𝑣differential-d𝜎differential-d𝑣\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
    0dσeσw0e(λ+σ/2)vGϑ(v)dvabsentsubscriptsuperscript0differential-d𝜎superscript𝑒𝜎𝑤subscriptsuperscript0superscript𝑒𝜆𝜎2𝑣subscript𝐺italic-ϑ𝑣differential-d𝑣\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
    =0eσwdσlog(λ+σ/2)ϑ+γ,absentsubscriptsuperscript0superscript𝑒𝜎𝑤d𝜎𝜆𝜎2italic-ϑ𝛾\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>eϑγ𝜆superscript𝑒italic-ϑ𝛾\lambda>e^{\vartheta-\gamma}italic_λ > italic_e start_POSTSUPERSCRIPT italic_ϑ - italic_γ end_POSTSUPERSCRIPT.To conclude, we choose w:=ur+12ur1assign𝑤subscript𝑢𝑟12subscript𝑢𝑟1w:=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,,m𝑟2𝑚r=2,...,mitalic_r = 2 , … , italic_m.∎

The next step is to integrate over the u𝑢uitalic_u variables in (3.6). Our approach here isinspired by [CZ23].However, some details are rather different as we make use of the multiplier λ𝜆\lambdaitalic_λ andwe also take into account the specifics of the criticality of the Critical 2d SHF.

To start with we define:

fλ(w):=w2Fλ(v)dv=01σeσwe2σlog(λ+σ/2)(ϑγ)dσ.assignsubscript𝑓𝜆𝑤subscriptsuperscript2𝑤subscript𝐹𝜆𝑣differential-d𝑣subscriptsuperscript01𝜎superscript𝑒𝜎𝑤superscript𝑒2𝜎𝜆𝜎2italic-ϑ𝛾differential-d𝜎f_{\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 fλ=Fλ0subscriptsuperscript𝑓𝜆subscript𝐹𝜆0f^{\prime}_{\lambda}=-F_{\lambda}\leq 0italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT = - italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ≤ 0 on (0,2]02(0,2]( 0 , 2 ], as F𝐹Fitalic_F is non-negative, thus, f𝑓fitalic_f is non-increasing.We have the following Lemma:

Lemma 3.4.

There exists C>0𝐶0C>0italic_C > 0 such that for all λ>(e2(ϑγ)1)𝜆superscript𝑒2italic-ϑ𝛾1\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)𝑤01w\in(0,1)italic_w ∈ ( 0 , 1 ) we have:

02Fλ(u+w)fλ(u)jdu=0j+1j!(j+1)!(4logλ)fλ(2w)j+1.superscriptsubscript02subscript𝐹𝜆𝑢𝑤subscript𝑓𝜆superscript𝑢𝑗differential-d𝑢subscriptsuperscript𝑗10𝑗𝑗1superscript4𝜆subscript𝑓𝜆superscript2𝑤𝑗1\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 Fλsubscript𝐹𝜆F_{\lambda}italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT and noting that for λ>e2(ϑγ)𝜆superscript𝑒2italic-ϑ𝛾\lambda>\text{$e^{2(\vartheta-\gamma)}$}italic_λ > italic_e start_POSTSUPERSCRIPT 2 ( italic_ϑ - italic_γ ) end_POSTSUPERSCRIPTand u0𝑢0u\geq 0italic_u ≥ 0:

    Fλ(u+w)subscript𝐹𝜆𝑢𝑤\displaystyle F_{\lambda}(u+w)italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_u + italic_w )Fλ(w)=0eσwdσlog(λ+σ/2)(ϑγ)20eσwlogλdσ=2wlogλ.absentsubscript𝐹𝜆𝑤subscriptsuperscript0superscript𝑒𝜎𝑤d𝜎𝜆𝜎2italic-ϑ𝛾2subscriptsuperscript0superscript𝑒𝜎𝑤𝜆differential-d𝜎2𝑤𝜆\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 02w()dusuperscriptsubscript02𝑤differential-d𝑢\int_{0}^{2w}(\cdots)\mathrm{d}u∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 italic_w end_POSTSUPERSCRIPT ( ⋯ ) roman_d italic_uand 2w2()dusuperscriptsubscript2𝑤2differential-d𝑢\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 I𝐼Iitalic_I and II𝐼𝐼IIitalic_I italic_I, respectively.We start by estimating integral I𝐼Iitalic_I. By (3.13) we have,

    I=02wFλ(u+w)fλ(u)jdu2wlogλ02wfλ(u)jdu𝐼subscriptsuperscript2𝑤0subscript𝐹𝜆𝑢𝑤subscript𝑓𝜆superscript𝑢𝑗differential-d𝑢2𝑤𝜆subscriptsuperscript2𝑤0subscript𝑓𝜆superscript𝑢𝑗differential-d𝑢\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,

    02wfλ(u)j𝑑u=2wfλ(2w)jj02wufλ(u)fλ(u)j1du2wfλ(2w)j+j2logλ02wfλ(u)j1du,subscriptsuperscript2𝑤0subscript𝑓𝜆superscript𝑢𝑗differential-d𝑢2𝑤subscript𝑓𝜆superscript2𝑤𝑗𝑗subscriptsuperscript2𝑤0𝑢subscriptsuperscript𝑓𝜆𝑢subscript𝑓𝜆superscript𝑢𝑗1differential-d𝑢2𝑤subscript𝑓𝜆superscript2𝑤𝑗𝑗2𝜆subscriptsuperscript2𝑤0subscript𝑓𝜆superscript𝑢𝑗1differential-d𝑢\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 ufλ(u)=uFλ(u)2logλ𝑢subscript𝑓𝜆superscript𝑢𝑢subscript𝐹𝜆𝑢2𝜆-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 j1𝑗1j\geq 1italic_j ≥ 1,

    02wfλ(u)jdu2wi=0jj!(ji)!(2logλ)ifλ(2w)ji,subscriptsuperscript2𝑤0subscript𝑓𝜆superscript𝑢𝑗differential-d𝑢2𝑤subscriptsuperscript𝑗𝑖0𝑗𝑗𝑖superscript2𝜆𝑖subscript𝑓𝜆superscript2𝑤𝑗𝑖\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

    Ii=0jj!(ji)!(4logλ)i+1fλ(2w)ji.𝐼subscriptsuperscript𝑗𝑖0𝑗𝑗𝑖superscript4𝜆𝑖1subscript𝑓𝜆superscript2𝑤𝑗𝑖\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, II𝐼𝐼IIitalic_I italic_I is estimated as:

    II:=2w2Fλ(u+w)fλ(u)jdu2w2Fλ(u)fλ(u)jdu=1j+1fλ(2w)j+1,assign𝐼𝐼superscriptsubscript2𝑤2subscript𝐹𝜆𝑢𝑤subscript𝑓𝜆superscript𝑢𝑗differential-d𝑢subscriptsuperscript22𝑤subscript𝐹𝜆𝑢subscript𝑓𝜆superscript𝑢𝑗differential-d𝑢1𝑗1subscript𝑓𝜆superscript2𝑤𝑗1\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 F𝐹Fitalic_F and the fact that f=Fsuperscript𝑓𝐹f^{\prime}=-Fitalic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = - italic_F. This completes the proof.∎

Lemma 3.5.

Fix m2𝑚2m\geq 2italic_m ≥ 2. For all 1km11𝑘𝑚11\leq k\leq m-11 ≤ italic_k ≤ italic_m - 1 and i=1mkui2subscriptsuperscript𝑚𝑘𝑖1subscript𝑢𝑖2\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 0ui20subscript𝑢𝑖20\leq u_{i}\leq 20 ≤ italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 2:

∫⋯∫i=mk+1mui2r=mk+1mFλ(ur+ur12)duri=0kcik(ki)!(4logλ)ifλ(umk)kisubscriptmultiple-integralsubscriptsuperscript𝑚𝑖𝑚𝑘1subscript𝑢𝑖2subscriptsuperscriptproduct𝑚𝑟𝑚𝑘1subscript𝐹𝜆subscript𝑢𝑟subscript𝑢𝑟12dsubscript𝑢𝑟subscriptsuperscript𝑘𝑖0subscriptsuperscript𝑐𝑘𝑖𝑘𝑖superscript4𝜆𝑖subscript𝑓𝜆superscriptsubscript𝑢𝑚𝑘𝑘𝑖\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 ciksubscriptsuperscript𝑐𝑘𝑖c^{k}_{i}italic_c start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are combinatorial coefficients defined inductively by

c00subscriptsuperscript𝑐00\displaystyle c^{0}_{0}italic_c start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT=1;cik=0fori>kandcik+1=j=0icjkforik+1.formulae-sequenceabsent1formulae-sequencesubscriptsuperscript𝑐𝑘𝑖0fori>kandsubscriptsuperscript𝑐𝑘1𝑖subscriptsuperscript𝑖𝑗0subscriptsuperscript𝑐𝑘𝑗forik+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=1𝑘1k=1italic_k = 1, the statement follows from Lemma 3.4 for j=0𝑗0j=0italic_j = 0 and w=ur12𝑤subscript𝑢𝑟12w=\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 k𝑘kitalic_k such that 1km21𝑘