Liouville dynamical percolation

• 成果类型: Article
• 署名作者: Garban, Christophe; Holden, Nina; Sepulveda, Avelio; Sun, Xin
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Institut Universitaire de France; Swiss Federal Institutes of Technology Domain; ETH Zurich; Universidad de Chile; University of Pennsylvania
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01057-1
• 发表日期: 2021
• 页码: 621-678
• 关键词: quantum-gravity; SCALING LIMITS; exceptional times; trees; plane

摘要

We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a gamma-Liouville measure. The homogeneous counterpart of this process describes the scaling limit of discrete dynamical percolation on the rescaled triangular lattice. Our focus here is to study the same limiting dynamics, but where the speed of microscopic updates is highly inhomogeneous in space and is driven by the gamma-Liouville measure associated with a two-dimensional log-correlated field h. Roughly speaking, this continuum percolation process evolves very rapidly where the field h is high and barely moves where the field h is low. Our main results can be summarized as follows. First, we build this inhomogeneous dynamical percolation, which we call gamma- Liouville dynamical percolation (LDP), by taking the scaling limit of the associated process on the triangular lattice. We work with three different regimes each requiring different tools: gamma is an element of [0, 2 - root 5/2), gamma is an element of [2 - root 5/2, root 3/2), and gamma is an element of (root 3/2, 2). When gamma < root 3/2, we prove that gamma-LDP is mixing in the Schramm-Smirnov space as t -> infinity, quenched in the log-correlated field h. On the contrary, when gamma > root 3/2 the process is frozen in time. The ergodicity result is a crucial piece of the Cardy embedding project of the second and fourth coauthors, where LDP for gamma = root 1/6 is used to study the scaling limit of a variant of dynamical percolation on uniform triangulations. When gamma < root 3/4, we obtain quantitative bounds on the mixing of quad crossing events.