Tightness for random walks driven by the two-dimensional Gaussian free field at high temperature

TL;DR

This paper investigates the behavior of random walks on two-dimensional Gaussian free field-driven networks at high temperature, establishing tightness as the mesh size approaches zero through analysis of effective resistances.

Contribution

It introduces a novel analysis of random walks in Gaussian free field environments, proving tightness at high temperature as the network becomes finer.

Findings

Proves tightness of random walks at high temperature

Analyzes effective resistances in Gaussian free field environments

Connects effective resistances to random walk behavior

Abstract

We study random walks in random environments generated by the two-dimensional Gaussian free field. More specifically, we consider a rescaled lattice with a small mesh size and view it as a random network where each edge is equipped with an electric resistance given by a regularization for the exponentiation of the Gaussian free field. We prove the tightness of random walks on such random networks at high temperature as the mesh size tends to 0. Our proof is based on a careful analysis of the (random) effective resistances as well as their connections to random walks.