TY - JOUR
T1 - From coalescing random walks on a torus to Kingman’s coalescent
AU - Beltrán, J.
AU - Chavez, E.
AU - Landim, C.
N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2019/12/1
Y1 - 2019/12/1
N2 - 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.
AB - 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.
KW - Interacting particle systems
KW - Kingman’s coalescent
KW - Markov chain model reduction
KW - Martingale problem
UR - http://www.scopus.com/inward/record.url?scp=85074586772&partnerID=8YFLogxK
UR - https://www.mendeley.com/catalogue/90c6b8b1-e5bf-3f12-a86d-47882b0c6eee/
U2 - 10.1007/s10955-019-02415-z
DO - 10.1007/s10955-019-02415-z
M3 - Article in a journal
AN - SCOPUS:85074586772
SN - 0022-4715
VL - 177
SP - 1172
EP - 1206
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 6
ER -