Liouville first passage percolation: the weight exponent is strictly less than 1 at high temperatures

Abstract

Let $\{{\eta }_{N,v}:v\in V_N\}$ be a discrete Gaussian free field in a two-dimensional box $V_N$ of side length $N$ with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e., the shortest path metric where each vertex is given a weight of $e^{\gamma {\eta }_{N,v}}$ for some $\gamma >0$. We show that for sufficiently small but fixed $\gamma >0$, the expected Liouville FPP distance between any pair of vertices is $O(N^{1-{\gamma }^2/10^3})$.

64 pages, 12 figures