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.
Categories: math.PR