2020/05/27 by Jian Ding, Ewain Gwynne, Ding, Jian +1 · 2 citations
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2005.13576
openalex publication_date 2020/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Liouville first passage percolation (LFPP) with parameter ξ>0 is the family of random distance functions \Dhε\ε>0 on the plane obtained by integrating eξhε along paths, where hε for ε>0 is a smooth mollification of the planar Gaussian free field. Previous work by Ding-Dubédat-Dunlap-Falconet and Gwynne-Miller has shown that there is a critical value ξcrit > 0 such that for ξ< ξcrit, LFPP converges under appropriate re-scaling to a random metric on the plane which induces the same topology as the Euclidean metric (the so-called γ-Liouville quantum gravity metric for γ= γ(ξ)∈ (0,2)). We show that for all ξ> 0, the LFPP metrics are tight with respect to the topology on lower semicontinuous functions. For ξ> ξcrit, every possible subsequential limit Dh is a metric on the plane which does not induce the Euclidean topology: rather, there is an uncountable, dense, Lebesgue measure-zero set of points z∈\mathbb C such that Dh(z,w) = ∞ for every w∈\mathbb C∖ \z\. We expect that these subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in (1,25).