METASTABLE BEHAVIOR OF WEAKLY MIXING MARKOV CHAINS: THE CASE OF REVERSIBLE, CRITICAL ZERO-RANGE PROCESSES

• 成果类型: Article
• 署名作者: Landim, C.; Marcondes, D.; Seo, I
• 署名单位: Instituto Nacional de Matematica Pura e Aplicada (IMPA); Universite de Rouen Normandie; Universidade de Sao Paulo; Seoul National University (SNU); Seoul National University (SNU)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1593
• 发表日期: 2023
• 页码: 157-227
• 关键词: diffusion; condensation; phase; MODEL; time

摘要

We present a general method to derive the metastable behavior of weakly mixing Markov chains. This approach is based on properties of the resolvent equations and can be applied to metastable dynamics, which do not satisfy the As an application, we study the metastable behavior of critical zero-range processes. Let r : S x S -> R+ be the jump rates of an irreducible random walk on a finite set S, reversible with respect to the uniform measure. For alpha > 0, let g : N -> R+ be given by g(0) = 0, g(1) = 1, g(k) = [k/(k - 1)]alpha, k > 2. Consider a zero-range process on S in which a particle jumps from a site x, occupied by k particles, to a site y at rate g(k)r(x, y). For alpha > 1, in the stationary state, as the total number of particles, represented by N, tends to infinity, all particles but a negligible number accumulate at one single site. This phenomenon is called condensation. Since condensation occurs if and only if alpha > 1, we call the case alpha =1 critical. By applying the general method established in the first part of the article to the critical case, we show that the site, which concentrates almost all particles, evolves in the time-scale N2 log N as a random walk on S whose transition rates are proportional to the capacities of the underlying random walk.