Maximum of the Ginzburg-Landau fields

TL;DR

This paper investigates the local distribution and extreme value behavior of the two-dimensional Ginzburg-Landau field with convex potential, extending known results for the Gaussian free field to a broader class of models.

Contribution

It proves Gaussian tail behavior for local marginals and characterizes the maximum and high points, generalizing previous results from the Gaussian free field to Ginzburg-Landau fields.

Findings

Local marginals have Gaussian tail distributions.

Characterization of the maximum and high points of the field.

Extension of Gaussian free field results to Ginzburg-Landau models.

Abstract

We study two dimensional massless field in a box with potential $V\left( \nabla \phi \left( \cdot \right) \right) $ and zero boundary condition, where $V$ is any symmetric and uniformly convex function. Naddaf-Spencer and Miller proved the macroscopic averages of this field converge to a continuum Gaussian free field. In this paper we prove the distribution of local marginal $\phi \left( x\right) $, for any $x$ in the bulk, has a Gaussian tail. We further characterize the leading order of the maximum and dimension of high points of this field, thus generalize the results of Bolthausen-Deuschel-Giacomin and Daviaud for the discrete Gaussian free field.