Extreme value statistics of 2d Gaussian Free Field: effect of finite domains

TL;DR

This paper investigates the minima statistics of the 2d Gaussian Free Field on finite domains, deriving free energy distributions and exploring duality and freezing phenomena with numerical validation.

Contribution

It provides exact calculations of free energy distributions for the 2d GFF on finite domains and demonstrates the duality property and freezing scenario predictions.

Findings

Free energy distributions satisfy duality property.

Minima distribution predicted by freezing scenario.

Numerical tests confirm theoretical predictions.

Abstract

We study minima statistics of the 2d Gaussian Free Field on circles in the unit disk with Dirichlet boundary condition. Free energy distributions of the associated Random Energy models are exactly calculated in the high temperature phase, and shown to satisfy the duality property, which enables us to predict the minima distribution by assuming the freezing scenario. Numerical tests are provided. Related questions concerning the GFF on a sphere are also considered.