Almost Sure Convergence of Liouville First Passage Percolation

2023/09/14 by Charles Devlin VI, VI, Charles Devlin
Computer Science · Mathematics · #60D05 #Bayesian Methods and Mixture Models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2309.08001

openalex publication_date 2023/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Related

• Tightness of supercritical Liouville first passage percolation
• Up-to-constants comparison of Liouville first passage percolation and Liouville quantum gravity
• The distance exponent for Liouville first passage percolation is positive
• Tightness of Liouville first passage percolation for γ∈ (0,2)
• Existence and uniqueness of the Liouville quantum gravity metric for γ∈ (0,2)