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

Jian Ding; Fuxi Zhang

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arXiv · 1610.02766

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

Jian Ding·Fuxi Zhang

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Abstract

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

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Jian Ding, Fuxi Zhang. 2019-03-16. Liouville first passage percolation: geodesic length exponent is strictly larger than 1 at high temperatures. https://arxiv.org/abs/1610.02766

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