The maximum of the four-dimensional membrane model

Abstract

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.

A limit law for the most favorite point of simple random walk on a regular tree

“…Randomly shifted Gumbel limit laws for centered maxima have been encountered in a number of contexts. These include Branching Brownian Motion (Bramson [20,21]) and critical Branching Random Walks (Aïdekon [6], Bramson, Ding and Zeitouni [22]) as well as the two-dimensional discrete GFF (Bramson, Ding and Zeitouni [23], Biskup and Louidor [16][17][18]) and other logarithmically correlated processes (e.g., Madaule [43], Ding, Roy and Zeitouni [31], Arguin and Oumet [11], Schweiger [44], Fels and Hartung [35]) including the local time for our simple random walk on T n run for times comparable with the cover time (Abe [2]).…”

Thermodynamic and Scaling Limits of the non-Gaussian Membrane Model

“…For pinning in d ≥ 4, some recent results are given in [18], and pinning in d = 2, 3 is not well understood. The behavior of the maximum height of the membrane for the critical dimension d = 4 was addressed in [19] and for d ≥ 5 in [6]. These results are all for the Gaussian model.…”

Optimal-Order Finite Difference Approximation of Generalized Solutions to the Biharmonic Equation in a Cube

“…In fact, the analysis pursued here was motivated by recent work by Müller and Schweiger [MS19], where estimates for the Green's function of the discrete bilaplacian on squares and cubes in two and three dimensions were proved. Very recently, Schweiger [Sch19] explored the behavior of the maximum of the solution to the four-dimensional membrane model; for that purpose the estimates from [MS19] are not sharp enough, but methods similar to those in the present paper can be employed to relate the Green's function of the discrete bilaplacian with its continuous counterpart and thereby to obtain the required bounds.…”

“…Similarly, one can relax the assumption s > 5 2 by replacing T h,2,...,2 with a stronger mollification operator. We will not pursue these alternatives here; see however [Sch19] for a version of Theorem 1.2 with s < 5 2 . 1.2.…”