EIGENVALUE VERSUS PERIMETER IN A SHAPE THEOREM FOR SELF-INTERACTING RANDOM WALKS

• 成果类型: Article
• 署名作者: Biskup, Marek; Procaccia, Eviatar B.
• 署名单位: University of California System; University of California Los Angeles; Charles University Prague; Texas A&M University System; Texas A&M University College Station
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1307
• 发表日期: 2018
• 页码: 340-377
• 关键词: 1st passage percolation; poissonian obstacles; wulff construction; brownian-motion; MODEL; confinement; crystal

摘要

We study paths of time-length t of a continuous-time random walk on Z(2) subject to self-interaction that depends on the geometry of the walk range and a collection of random, uniformly positive and finite edge weights. The interaction enters through a Gibbs weight at inverse temperature beta; the energy is the total sum of the edge weights for edges on the outer boundary of the range. For edge weights sampled from a translation-invariant, ergodic law, we prove that the range boundary condensates around an asymptotic shape in the limit t -> infinity followed by beta -> infinity. The limit shape is a minimizer (unique, modulo translates) of the sum of the principal harmonic frequency of the domain and the perimeter with respect to the first-passage percolation norm derived from (the law of) the edge weights. A dense subset of all norms in R-2, and thus a large variety of shapes, arise from the class of weight distributions to which our proofs apply.