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 language | English |
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Pages (from-to) | 1172-1206 |
Number of pages | 35 |
Journal | Journal of Statistical Physics |
Volume | 177 |
Issue number | 6 |
DOIs | |
State | Published - 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