From coalescing random walks on a torus to Kingman’s coalescent

J. Beltrán, E. Chavez, C. Landim

Research output: Contribution to journalArticle in a journalpeer-review

2 Scopus citations

Abstract

Let TNd, d≥ 2 , be the discrete d-dimensional torus with Nd points. Place a particle at each site of TNd and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they coalesce into one. Denote by CN the first time the set of particles is reduced to a singleton. Cox (Ann Probab 17:1333–1366, 1989) proved the existence of a time-scale θN for which CN/ θN converges to the sum of independent exponential random variables. Denote by ZtN the total number of particles at time t. We prove that the sequence of Markov chains (ZtθNN)t≥0 converges to the total number of partitions in Kingman’s coalescent.

Original languageEnglish
Pages (from-to)1172-1206
Number of pages35
JournalJournal of Statistical Physics
Volume177
Issue number6
DOIs
StatePublished - 1 Dec 2019

Bibliographical note

Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Interacting particle systems
  • Kingman’s coalescent
  • Markov chain model reduction
  • Martingale problem

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