arXiv subjects

Florian Schweiger

Publications and source records attributed to Florian Schweiger.

12 recordsLinked to original sources

arXiv2026

Gradient Gibbs measures with non-convex potentials and the universality class of the Gaussian Free Field

We study a general class of gradient interface models with Hamiltonian $H=β\sum V(\nablaϕ)$, $β>0$, assuming essentially that the potential $V$ is even, $V'(s)\ge αs$ on $[0,\infty)$ for some $α>0$, and $-M\leq V''\le C$. We establish a Helffer-Sjöstrand representation for these models, and use it to prove that their scaling limits are Gaussian Free Fields (GFFs), and that their covariances decay at the same rate as the GFF. This extends results for strictly convex potentials to a large class of non-convex potentials and to arbitrary temperatures. Additionally, we prove Brascamp-Lieb and dimension-free Poincaré inequalities for the models. We obtain these results by representing the interface as a mixture of gradient interface models with strictly convex potentials, extending an idea by Biskup-Spohn who had considered mixtures of Gaussians at moderate inverse temperature $β=1$. The construction of such a representation is one of the key new contributions of this work.

math.PR↗

arXiv2025

Quantitative delocalization for solid-on-solid models at high temperature and arbitrary tilt

We study a family of integer-valued random interface models on the two-dimensional square lattice that include the solid-on-solid model and more generally $p$-SOS models for $0<p\le2$, and prove that at sufficiently high temperature the interface is delocalized logarithmically uniformly in the boundary data. Fröhlich and Spencer had studied the analogous problem with free boundary data, and our proof is based on their multi-scale argument, with various technical improvements.

math.PR↗

arXiv2025

Scaling limits for fractional polyharmonic Gaussian fields

This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covariance operator is given by the inverse fractional Laplacian $(-Δ)^{-s}$ (where, in particular, we include the case $s >1$). We define a lattice discretization of these fields and show that their scaling limits -- with respect to the optimal Besov space topology (up to an endpoint case) -- are the original continuous fields. As a byproduct, in dimension $d<2s$, we prove the convergence in distribution of the maximum of the fields. A key tool in the proof is a sharp error estimate for the natural finite difference scheme for $(-Δ)^s$ under minimal regularity assumptions, which is also of independent interest.

math.PR↗

arXiv2024

Tightness of the maximum of Ginzburg-Landau fields

We consider the discrete Ginzburg-Landau field with potential satisfying a uniform convexity condition, in the critical dimension $d=2$, and prove that its maximum over boxes of sidelength $N$, centered by an explicit $N$-dependent centering, is tight.

math.PR↗

arXiv2023

Finite range decompositions of Gaussian fields with applications to level-set percolation

In a recent work (arXiv:2206.10724), Muirhead has studied level-set percolation of (discrete or continuous) Gaussian fields, and has shown sharpness of the associated phase transition under the assumption that the field has a certain multiscale white noise decomposition, a variant of a finite-range decomposition. We show that a large class of Gaussian fields have such a white noise decomposition with optimal decay parameter. Examples include the discrete Gaussian free field, the discrete membrane model, and the mollified continuous Gaussian free field. This answers various questions from Muirhead's paper. Our construction of the white-noise decomposition is a refinement of Bauerschmidt's construction of finite-range decomposition. In the continuous setting our construction is very similar to Bauerschmidt's, while in the discrete setting several new ideas are needed, including the use of a result by Pólya and Szegő on polynomials that take positive values on the positive real line.

math.PR↗

arXiv2023

Asymptotics of the $p$-capacity in the critical regime

In this note, we are interested in the asymptotics as $n\to\infty$ of the $p$-capacity between the origin and the set $nB$, where $B$ is the boundary of the unit ball of the lattice $\mathbb Z^d$. The $p$-capacity is defined as the minimum of the Dirichlet energy $\frac{1}{2}\sum_{x\in \mathbb Z^d} \sum_{y\sim x} |f(x)-f(y)|^{p}$ with $f$ subject to the boundary conditions $f(0)=0$ and $f\geq 1$ on $nB$. This variational problem has arisen in particular in the study of large deviations for first passage percolation. For $p d$ the capacity vanishes polynomially fast. The present paper deals with the case $p=d$, for which we prove that the $p$-capacity vanishes as $c_d (\log n)^{-d+1}$ with an explicit constant $c_d$.

math.AP↗

arXiv2022

The maximum of log-correlated Gaussian fields in random environments

We study the distribution of the maximum of a large class of Gaussian fields indexed by a box $V_N\subset Z^d$ and possessing logarithmic correlations up to local defects that are sufficiently rare. Under appropriate assumptions that generalize those in Ding, Roy and Zeitouni (Annals Probab. (45) 2017, 3886-3928), 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 field in i.i.d. bounded conductances, fall under the assumptions of our general theorem.

math.PR↗

arXiv2020

Pinning for the critical and supercritical membrane model

The membrane model is a Gaussian interface model with a Hamiltonian involving second derivatives of the interface height. We consider the model in dimension $\mathsf{d}\ge4$ under the influence of $δ$-pinning of strength $\varepsilon$. It is known that this pinning potential manages to localize the interface for any $\varepsilon>0$. We refine this result by establishing the $\varepsilon$-dependence of the variance and of the exponential decay rate of the covariances for small $\varepsilon$ (similar to the corresponding results for the discrete Gaussian free field by Bolthausen-Velenik). We also show the existence of a thermodynamic limit of the field. These conclusions improve upon earlier works by Bolthausen-Cipriani-Kurt and by Sakagawa. The problem has similarities to the homogenization of elliptic operators in randomly perforated domains, and our proof takes inspiration from this connection. The main new ideas are a correlation inequality for the set of pinned points, and a probabilistic Widman hole filler argument which relies on a discrete multipolar Hardy-Rellich inequality and on a multi-scale argument to construct suitable test functions.

math.PR↗

arXiv2019

The maximum of the four-dimensional membrane model

We show that the centred maximum of the four-dimensional membrane model on a box of sidelength $N$ converges in distribution. To do so we use a criterion of Ding, Roy and Zeitouni and prove sharp estimates for the Green's function of the discrete Bilaplacian. These estimates are the main contribution of this work and might also be of independent interest. To derive them we use estimates for the approximation quality of finite difference schemes as well as results for the Green's function of the continuous Bilaplacian.

math.PR↗

arXiv2019

Optimal order finite difference approximation of generalized solutions to the biharmonic equation in a cube

We prove an optimal order error bound in the discrete $H^2(Ω)$ norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in $n$ space dimensions, with $n \in \{2,\dots,7\}$, whose generalized solution belongs to the Sobolev space $H^s(Ω) \cap H^2_0(Ω)$, for $\frac{1}{2} \max(5,n) < s \leq 4$, where $Ω= (0,1)^n$. The result extends the range of the Sobolev index $s$ in the best convergence results currently available in the literature to the maximal range admitted by the Sobolev embedding of $H^s(Ω)$ into $C(\overlineΩ)$ in $n$ space dimensions.

math.NA↗

arXiv2018

Probability to be positive for the membrane model in dimensions 2 and 3

We consider the membrane model on a box $V_N\subset \mathbb{Z}^n$ of size $(2N+1)^n$ with zero boundary condition in the subcritical dimensions $n=2$ and $n=3$. We show optimal estimates for the probability that the field is positive in a subset $D_N$ of $V_N$. In particular we obtain for $D_N=V_N$ that the probability to be positive on the entire domain is exponentially small and the rate is of the order of the surface area $N^{n-1}$.

math.PR↗

arXiv2017

Estimates for the Green's function of the discrete bilaplacian in dimensions 2 and 3

We prove estimates for the Green's function of the discrete bilaplacian in squares and cubes in two and three dimensions which are optimal except possibly near the corners of the square and the edges and corners of the cube. The main idea is to transfer estimates for the continuous bilaplacian using a new discrete compactness argument and a discrete version of the Cacciopoli (or reverse Poincaré) inequality. One application that we have in mind is the study of entropic repulsion for the membrane model from statistical physics.

math-ph↗