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March 2019 Liouville first-passage percolation: Subsequential scaling limits at high temperature
Jian Ding, Alexander Dunlap
Ann. Probab. 47(2): 690-742 (March 2019). DOI: 10.1214/18-AOP1267

Abstract

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-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.

Citation

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Jian Ding. Alexander Dunlap. "Liouville first-passage percolation: Subsequential scaling limits at high temperature." Ann. Probab. 47 (2) 690 - 742, March 2019. https://doi.org/10.1214/18-AOP1267

Information

Received: 1 September 2016; Revised: 1 November 2017; Published: March 2019
First available in Project Euclid: 26 February 2019

zbMATH: 07053554
MathSciNet: MR3916932
Digital Object Identifier: 10.1214/18-AOP1267

Subjects:
Primary: 60K35
Secondary: 60B43, 60G60

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 2 • March 2019
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