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

Let $\{Y_{\mathfrak{B}}(x): x ∈ \mathfrak{B}\}$ 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 $e^{γ Y_{\mathfrak{B}}(x)}$ 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 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-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.

Duke Scholars

Author Alexander J Dunlap Mathematics