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From Coalescing Random Walks on a Torus to Kingman’s Coalescent


Let \({\mathbb T}^d_N\), \(d\ge 2\), be the discrete d-dimensional torus with \(N^d\) points. Place a particle at each site of \({\mathbb T}^d_N\) and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they coalesce into one. Denote by \(C_N\) 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 \(\theta _N\) for which \(C_N/\theta _N\) converges to the sum of independent exponential random variables. Denote by \(Z^N_t\) the total number of particles at time t. We prove that the sequence of Markov chains \((Z^N_{t\theta _N})_{t\ge 0}\) converges to the total number of partitions in Kingman’s coalescent.

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Communicated by Abhishek Dhar.

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Beltrán, J., Chavez, E. & Landim, C. From Coalescing Random Walks on a Torus to Kingman’s Coalescent. J Stat Phys 177, 1172–1206 (2019).

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

Mathematics Subject Classification

  • 82C22
  • 60K35
  • 60F99