The Annals of Probability

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

Jian Ding and Alexander Dunlap

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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.

Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 690-742.

Dates
Received: September 2016
Revised: November 2017
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1551171635

Digital Object Identifier
doi:10.1214/18-AOP1267

Mathematical Reviews number (MathSciNet)
MR3916932

Zentralblatt MATH identifier
07053554

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G60: Random fields 60B43

Keywords
Liouville first-passage percolation discrete Gaussian free field Russo–Seymour–Welsh method Liouville quantum gravity

Citation

Ding, Jian; Dunlap, Alexander. Liouville first-passage percolation: Subsequential scaling limits at high temperature. Ann. Probab. 47 (2019), no. 2, 690--742. doi:10.1214/18-AOP1267. https://projecteuclid.org/euclid.aop/1551171635


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