Almost Sure Convergence of Liouville First Passage Percolation

TL;DR

This paper proves that Liouville first passage percolation metrics converge almost surely to a limiting metric, confirming a conjecture and advancing understanding of Liouville quantum gravity models.

Contribution

It establishes almost sure convergence of LFPP to the limiting metric, answering a question posed by Gwynne and Miller (2019).

Findings

Almost sure convergence of LFPP metrics.

Confirmation of the limiting metric's existence.

Advancement in Liouville quantum gravity theory.

Abstract

Liouville first passage percolation (LFPP) with parameter $\xi > 0$ is the family of random distance functions (metrics) $(D_h^{\epsilon})_{\epsilon > 0}$ on $\mathbb{C}$ obtained heuristically by integrating $e^{\xi h}$ along paths, where $h$ is a variant of the Gaussian free field. There is a critical value $\xi_{\text{crit}} \approx 0.41$ such that for $\xi \in (0, \xi_{\text{crit}})$, appropriately rescaled LFPP converges in probability uniformly on compact subsets of $\mathbb{C}$ to a limiting metric $D_h$ on $\gamma$-Liouville quantum gravity with $\gamma = \gamma(\xi) \in (0,2)$. We show that the convergence is almost sure, giving an affirmative answer to a question posed by Gwynne and Miller (2019).