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Let $$\{\eta (v): v\in V_N\}$${η(v):v∈VN} be a discrete Gaussian free field in a two-dimensional box $$V_N$$VN 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 $$e^{\gamma \eta (v)}$$eγη(v) for some $$\gamma >0$$γ>0. We show that for sufficiently small but fixed $$\gamma >0$$γ>0, with probability tending to 1 as $$N\rightarrow \infty $$N→∞, all geodesics between vertices of macroscopic Euclidean distances simultaneously have (the conjecturally unique) length exponent strictly larger than 1.

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