## 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.

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- Statistics & Probability
- 4905 Statistics
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### Citation

*Annals of Probability*,

*47*(2), 690–742. https://doi.org/10.1214/18-AOP1267

*Annals of Probability*47, no. 2 (January 1, 2019): 690–742. https://doi.org/10.1214/18-AOP1267.

*Annals of Probability*, vol. 47, no. 2, Jan. 2019, pp. 690–742.

*Manual*, doi:10.1214/18-AOP1267.

## Published In

## DOI

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Statistics & Probability
- 4905 Statistics
- 4904 Pure mathematics
- 0104 Statistics
- 0101 Pure Mathematics