The distance exponent for Liouville first passage percolation is positive

TL;DR

This paper proves that the distance exponent in Liouville first passage percolation (LFPP) is positive for all parameters, establishing a key property for understanding the metric's scaling limits and implications in Liouville quantum gravity.

Contribution

It demonstrates that the distance exponent in discrete LFPP is strictly positive for all positive parameters, a crucial step for analyzing its scaling limits and quantum gravity connections.

Findings

LFPP distance between boundaries grows at least exponentially with size

Distance exponent is strictly positive for all $\xi > 0$

Supports existence of non-trivial scaling limits for LFPP

Abstract

Discrete Liouville first passage percolation (LFPP) with parameter $\xi > 0$ is the random metric on a sub-graph of $\mathbb Z^2$ obtained by assigning each vertex $z$ a weight of $e^{\xi h(z)}$, where $h$ is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all $\xi > 0$. More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size $2^n$ is typically at least $2^{\alpha n}$ for an exponent $\alpha > 0$ depending on $\xi$. This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all $\xi > 0$ and also has theoretical implications for the study of distances in Liouville quantum gravity.