The maximum of log‐correlated Gaussian fields in random environment

Schweiger, Florian; Zeitouni, Ofer

doi:10.1002/cpa.22181

Comm Pure Appl Math

2023

DOI: 10.1002/cpa.22181

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The maximum of log‐correlated Gaussian fields in random environment

Florian Schweiger,

Ofer Zeitouni

Abstract: We study the distribution of the maximum of a large class of Gaussian fields indexed by a box and possessing logarithmic correlations up to local defects that are sufficiently rare. Under appropriate assumptions that generalize those in Ding et al., we show that asymptotically, the centered maximum of the field has a randomly‐shifted Gumbel distribution. We prove that the two dimensional Gaussian free field on a super‐critical bond percolation cluster with p close enough to 1, as well as the Gaussian free fie… Show more

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Cited by 2 publications

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Stochastic Homogenization of Gaussian Fields on Random Media

Chiarini

Ruszel

2023

Ann. Henri Poincaré

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In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $$\Xi ^{g,\textbf{a}} $$ Ξ g , a and bi-Laplacian fields $$\Xi ^{b,\textbf{a}}$$ Ξ b , a . They can be characterized as follows: for $$f=\delta $$ f = δ the solution u of $$\nabla \cdot \textbf{a} \nabla u =f$$ ∇ · a ∇ u = f , $$\textbf{a}$$ a is a uniformly elliptic random environment, is the covariance of $$\Xi ^{g,\textbf{a}}$$ Ξ g , a . When f is the white noise, the field $$\Xi ^{b,\textbf{a}}$$ Ξ b , a can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain $$D\subset \mathbb {R}^d$$ D ⊂ R d , or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator $$\Delta $$ Δ , we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator $${{\,\mathrm{\bar{\textbf{a}}}\,}}\Delta $$ a ¯ Δ , with constant $${{\,\mathrm{\bar{\textbf{a}}}\,}}$$ a ¯ depending on the law of the environment $$\textbf{a}$$ a . The proofs are based on the results found in Armstrong et al. (in: Grundlehren der mathematischen Wissenschaften, Springer International Publishing, Cham, 2019) and Gloria et al. (ESAIM Math Model Numer Anal 48(2):325-346, 2014).

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Stochastic Homogenization of Gaussian Fields on Random Media

Chiarini

Ruszel

2023

Ann. Henri Poincaré

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Multiscale Coupling and the Maximum of $${\mathcal {P}}(\phi )_2$$ Models on the Torus

Barashkov,

Gunaratnam,

Hofstetter

2023

Commun. Math. Phys.

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We establish a coupling between the $${\mathcal {P}}(\phi )_2$$ P ( ϕ ) 2 measure and the Gaussian free field on the two-dimensional unit torus at all spatial scales, quantified by probabilistic regularity estimates on the difference field. Our result includes the well-studied $$\phi ^4_2$$ ϕ 2 4 measure. The proof uses an exact correspondence between the Polchinski renormalisation group approach, which is used to define the coupling, and the Boué–Dupuis stochastic control representation for $${\mathcal {P}}(\phi )_2$$ P ( ϕ ) 2 . More precisely, we show that the difference field is obtained from a specific minimiser of the variational problem. This allows to transfer regularity estimates for the small-scales of minimisers, obtained using discrete harmonic analysis tools, to the difference field.As an application of the coupling, we prove that the maximum of the $${\mathcal {P}}(\phi )_2$$ P ( ϕ ) 2 field on the discretised torus with mesh-size $$\epsilon > 0$$ ϵ > 0 converges in distribution to a randomly shifted Gumbel distribution as $$\epsilon \rightarrow 0$$ ϵ → 0 .

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The maximum of log‐correlated Gaussian fields in random environment

Cited by 2 publications

References 50 publications

Stochastic Homogenization of Gaussian Fields on Random Media

Stochastic Homogenization of Gaussian Fields on Random Media

Multiscale Coupling and the Maximum of $${\mathcal {P}}(\phi )_2$$ Models on the Torus

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