Abstract
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 mixing conditions required in (J. Stat. Phys. 140 (2010) 1065–1114; J. Stat. Phys. 149 (2012) 598–618) or in Landim, Marcondes and Seo (2020).
As an application, we study the metastable behavior of critical zero-range processes. Let be the jump rates of an irreducible random walk on a finite set S, reversible with respect to the uniform measure. For , let be given by , , , . 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 . For , 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 , we call the case 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 as a random walk on S whose transition rates are proportional to the capacities of the underlying random walk.
Funding Statement
C. L. has been partially supported by FAPERJ CNE E-26/201.207/2014, by CNPq Bolsa de Produtividade em Pesquisa PQ 303538/2014-7, by ANR-15-CE40-0020-01 LSD of the French National Research Agency.
I.S. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2018R1C1B6006896 and No. 2017R1A5A1015626), the Samsung Science and Technology Foundation (Project Number SSTF-BA1901-03) and POSCO Science Fellowship of POSCO TJ Park Foundation.
D. M. has received financial support from CNPq during the development of this paper.
Acknowledgments
The authors wish to thank M. Loulakis and S. Grosskinsky for references on the Efron–Stein inequality.
Part of this work was done when the first two authors were at the Seoul National University. The warm hospitality is acknowledged.
Citation
C. Landim. D. Marcondes. I. Seo. "Metastable behavior of weakly mixing Markov chains: The case of reversible, critical zero-range processes." Ann. Probab. 51 (1) 157 - 227, January 2023. https://doi.org/10.1214/22-AOP1593
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