Let @math be a discrete Gaussian free field in a two-dimensional box @math of side length @math with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e., the shortest path metric where each vertex is given a weight of @math for some @math . We show that for sufficiently small but fixed @math , with probability tending to @math as @math , all geodesics between vertices of macroscopic Euclidean distances simultaneously have (the conjecturally unique) length exponent strictly larger than 1.
In @cite_4 the chemical distance (i.e., graph distance in the induced open cluster) for the percolation of level sets of two-dimensional DGFF was studied, and in particular [Theorem 1.1] DL16 implies that there exists a path of dimension 1 joining the two boundaries of an annulus which has Liouville FPP weight @math . This can be seen as a complement of our Theorem . In @cite_9 , a non-universality result was proved on the weight exponent for the geodesic among log-correlated Gaussian fields, which suggests the subtlety when attempting to estimate the weight exponent. We would like to point out that the proof of our main result Theorem is expected to be adaptable to general log-correlated Gaussian fields with @math -scale invariant kernels as in @cite_1 . However, in light of @cite_9 there seems to be no reason to expect that the geodesic dimension is universal among log-correlated fields.
@cite_1 @cite_9 @cite_4
2022385268 2964208310
In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of logcorrelated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables. 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( ) ), where N is the side length of the box and ( ) 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.