The distance exponent for Liouville first passage percolation is positive.

Discrete Liouville first passage percolation (LFPP) with parameter ξ > 0 is the random metric on a sub-graph of Z 2 obtained by assigning each vertex z a weight of e ξ h (z) , where h is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all ξ > 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 α n for an exponent α > 0 depending on ξ . This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all ξ > 0 and also has theoretical implications for the study of distances in Liouville quantum gravity. [ABSTRACT FROM AUTHOR]