Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields

We consider first passage percolation (FPP) where the vertex weight is given by the exponential of two-dimensional log-correlated Gaussian fields. Our work is motivated by understanding the discrete analog for the random metric associated with Liouville quantum gravity (LQG), which roughly corresponds to the exponential of a two-dimensional Gaussian free field (GFF). The particular focus of the present paper is an aspect of universality for such FPP among the family of log-correlated Gaussian fields. More precisely, we construct a family of log-correlated Gaussian fields, and show that the FPP distance between two typically sampled vertices (according to the LQG measure) is N^1+ O(ε ) , where N is the side length of the box and ε can be made arbitrarily small if we tune a certain parameter in our construction. That is, the exponents can be arbitrarily close to 1. Combined with a recent work of the first author and Goswami on an upper bound for this exponent when the underlying field is a GFF, our result implies that such exponent is not universal among the family of log-correlated Gaussian fields.