Liouville first-passage percolation: Subsequential scaling limits at high temperature.

Let {YB(x):x.B} be a discrete Gaussian free field in a two-dimensional box B of side length S with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex x is given a weight of e.YB(x) for some .>0. We show that for sufficiently small but fixed .>0, for any sequence of scales {Sk} there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov.Hausdorff sense to a random metric on the unit square in R2. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.