Let {Y B (v) : v ∈ B} be a discrete Gaussian free field in a two-dimensional box B of side length S with Dirichlet boundary conditions. We study the Liouville first-passage percolation, in which each vertex is given a weight of e γY B (v) for some γ > 0. We show that for sufficiently small but fixed γ > 0, for any sequence of scales {S k } there exists a subsequence along which the appropriately scaled Liouville FPP metric converges in the Gromov-Hausdorff sense to a random metric on the unit square in R 2 . In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.
In a recent work REF , it was shown that at high temperatures the appropriately normalized Liouville FPP converges subsequentially in the Gromov-Hausdorff sense to a random metric on the unit square, where all the (conjecturally unique) limiting metrics are homeomorphic to the Euclidean metric.
Liouville first-passage percolation: subsequential scaling limits at high temperature
arXiv: Probability
Mathematics