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

Abstract

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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References

  1. 1.

    Aldous, D., Fill, J.A.: Reversible Markov chains and random walks on graphs. http://www.stat.berkeley.edu/~aldous/RWG/book.html (2001)

  2. 2.

    Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys. 140, 1065–1114 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains II. J. Stat. Phys. 149, 598–618 (2012)

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, Y.T., Choi, J., Cox, J.T.: On the convergence of densities of finite voter models to the Wright-Fisher diffusion. Ann. Inst. H. Poincaré Probab. Stat. 52, 286–322 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Cooper, C., Frieze, A., Radzik, T.: Multiple random walks in random regular graphs. SIAM J. Discret. Math. 23, 1738–1761 (2009)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cox, J.T.: Coalescing random walks and voter model consensus times on the torus in \(Z^d\). Ann. Probab. 17, 1333–1366 (1989)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Durrett, R.: Some features of the spread of epidemics and information on a random graph. Proc. Nat. Acad. Sci. USA 107, 4491–4498 (2010)

    ADS  Article  Google Scholar 

  8. 8.

    Heuer, B., Sturm, A.: On spatial coalescents with multiple mergers in two dimensions. Theor. Popul. Biol. 87, 90–104 (2013)

    Article  Google Scholar 

  9. 9.

    Jara, M., Landim, C., Teixeira, A.: Quenched scaling limits of trap models. Ann. Probab. 39, 176–223 (2011)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kingman, J.F.C.: Coalescent. Stoch. Proc. Appl. 13, 235–248 (1982)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kolokoltsov, V.N.: Markov Processes, Semigroups and Generators. De Gruyter Studies in Mathematics, vol. 38. De Gruyter, Berlin (2011)

    Google Scholar 

  12. 12.

    Lawler, G.F.: Intersections of Random Walks. Modern Birkhäuser Classics. Birkhäuser, Basel (1991)

    Book  Google Scholar 

  13. 13.

    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  14. 14.

    Limic, V., Sturm, A.: The spatial \(\Lambda \)-coalescent. Elect. J. Probab. 11, 363–393 (2006)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Oliveira, R.I.: On the coalescence time of reversible random walks. Trans. Am. Math. Soc. 364, 2109–2128 (2012)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Oliveira, R.I.: Mean field conditions for coalescing random walks. Ann. Probab. 41, 3420–3461 (2013)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Grundlehren der mathematischen Wissenschaften, vol. 233. Springer, Berlin (1979)

    Google Scholar 

  18. 18.

    Zähle, I., Cox, J.T., Durrett, R.: The stepping stone model. II: genealogies and the infinite sites model. Ann. Appl. Probab. 15, 671–699 (2005)

    MathSciNet  Article  Google Scholar 

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Correspondence to C. Landim.

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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). https://doi.org/10.1007/s10955-019-02415-z

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Keywords

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

Mathematics Subject Classification

  • 82C22
  • 60K35
  • 60F99