Liouville first passage percolation: geodesic length exponent is strictly larger than 1 at high temperatures

Abstract

Let $$\{\eta (v): v\in V_N\}$$ 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 $$e^{\gamma \eta (v)}$$ for some $$\gamma >0$$ . We show that for sufficiently small but fixed $$\gamma >0$$ , with probability tending to 1 as $$N\rightarrow \infty $$ , all geodesics between vertices of macroscopic Euclidean distances simultaneously have (the conjecturally unique) length exponent strictly larger than 1.