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 \emph{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 $N^{1+ O(\epsilon)}$, where $N$ is the side length of the box and $\epsilon$ 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 \emph{not} universal among the family of log-correlated Gaussian fields.