Liouville first-passage percolation: subsequential scaling limits at high temperature

Let $\{Y_{\mathfrak{B}}(x):x\in \mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ 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 $e^{\gamma Y_{\mathfrak{B}}(x)}$ for some $\gamma >0$. We show that for sufficiently small but fixed $\gamma >0$, for any sequence of scales $\{S_k\}$ 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 ${\mathbf{R}}^2$. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-H"older-continuous homeomorphisms to the unit square with the Euclidean metric.