Liouville first-passage percolation: subsequential scaling limits at high temperature

TL;DR

This paper investigates the subsequential scaling limits of Liouville first-passage percolation in two dimensions, showing convergence to a random metric and homeomorphism to the Euclidean square at high temperature.

Contribution

It establishes the existence of subsequential scaling limits for Liouville FPP in 2D and characterizes their topological properties at high temperature.

Findings

Existence of subsequential Gromov–Hausdorff limits for Liouville FPP.

Scaling limits are homeomorphic to the Euclidean square.

Limits are conjecturally unique and bi-Hölder continuous.

Abstract

Let $\{Y_{\mathfrak{B}}(x)\,:\,x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{\gamma Y_{\mathfrak{B}}(x)}$ for some $\gamma>0$. We show that for sufficiently small but fixed $\gamma>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov--Hausdorff sense to a random metric on the unit square in $\mathbf{R}^{2}$. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-H\"older-continuous homeomorphisms to the unit square with the Euclidean metric.