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

Let $\{η_{N, v}: v\in V_N\}$ 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 $e^{γη_{N, v}}$ for some $γ>0$. We show that for sufficiently small but fixed $γ>0$, the expected Liouville FPP distance between any pair of vertices is $O(N^{1-γ^2/10^3})$.

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