The distance exponent for Liouville first passage percolation is positive

Abstract

Discrete Liouville first passage percolation (LFPP) with parameter xi > 0 is the random metric on a sub-graph of 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.