Maximum of the Ginzburg-Landau fields

Abstract

We study two dimensional massless field in a box with potential $V\left(\nabla \varphi \left(\cdot \right)\right)$ 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 $\varphi \left(x\right)$, 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.

References in corpus (2)

• Convergence of the centered maximum of log-correlated Gaussian fields
• Universality for SLE(4)

Cited by in corpus (4)

• Extrema of the two-dimensional Discrete Gaussian Free Field
• Maximum and coupling of the sine-Gordon field
• Critical Gaussian multiplicative chaos: a review
• Subsequential tightness of the maximum of two dimensional Ginzburg-Landau fields