The Annals of Probability

Mean field conditions for coalescing random walks

Roberto Imbuzeiro Oliveira

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Abstract

The main results in this paper are about the full coalescence time $\mathsf{C}$ of a system of coalescing random walks over a finite graph $G$. Letting $\mathsf{m}(G)$ denote the mean meeting time of two such walkers, we give sufficient conditions under which $\mathbf{E}[\mathsf{C}]\approx2\mathsf{m}(G)$ and $\mathsf{C}/\mathsf{m}(G)$ has approximately the same law as in the “mean field” setting of a large complete graph. One of our theorems is that mean field behavior occurs over all vertex-transitive graphs whose mixing times are much smaller than $\mathsf{m}(G)$; this nearly solves an open problem of Aldous and Fill and also generalizes results of Cox for discrete tori in $d\geq2$ dimensions. Other results apply to nonreversible walks and also generalize previous theorems of Durrett and Cooper et al. Slight extensions of these results apply to voter model consensus times, which are related to coalescing random walks via duality.

Our main proof ideas are a strengthening of the usual approximation of hitting times by exponential random variables, which give results for nonstationary initial states; and a new general set of conditions under which we can prove that the hitting time of a union of sets behaves like a minimum of independent exponentials. In particular, this will show that the first meeting time among $k$ random walkers has mean $\approx\mathsf{m}(G)/\bigl({\matrix{k\\2}}\bigr)$.

Article information

Source
Ann. Probab., Volume 41, Number 5 (2013), 3420-3461.

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1378991844

Digital Object Identifier
doi:10.1214/12-AOP813

Mathematical Reviews number (MathSciNet)
MR3127887

Zentralblatt MATH identifier
1285.60094

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces

Keywords
Coalescing random walks voter model hitting times exponential approximation

Citation

Oliveira, Roberto Imbuzeiro. Mean field conditions for coalescing random walks. Ann. Probab. 41 (2013), no. 5, 3420--3461. doi:10.1214/12-AOP813. https://projecteuclid.org/euclid.aop/1378991844


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References

  • [1] Aldous, D. (2010). Mixing times and hitting times. Available at http://www.stat.berkeley.edu/~aldous/Talks/slides.html.
  • [2] Aldous, D. and Fill, J. A. (2001). Reversible Markov chains and random walks on graphs. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • [3] Aldous, D. J. (1982). Markov chains with almost exponential hitting times. Stochastic Process. Appl. 13 305–310.
  • [4] Aldous, D. J. and Brown, M. (1992). Inequalities for rare events in time-reversible Markov chains. I. In Stochastic Inequalities (Seattle, WA, 1991). Institute of Mathematical Statistics Lecture Notes—Monograph Series 22 1–16. IMS, Hayward, CA.
  • [5] Benjamini, I. and Mossel, E. (2003). On the mixing time of a simple random walk on the super critical percolation cluster. Probab. Theory Related Fields 125 408–420.
  • [6] Cooper, C., Frieze, A. and Radzik, T. (2009). Multiple random walks in random regular graphs. SIAM J. Discrete Math. 23 1738–1761.
  • [7] Cox, J. T. (1989). Coalescing random walks and voter model consensus times on the torus in $\mathbb{Z}^{d}$. Ann. Probab. 17 1333–1366.
  • [8] Durrett, R. (2007). Random Graph Dynamics. Cambridge Univ. Press, Cambridge.
  • [9] Durrett, R. (2010). Some features of the spread of epidemics and information on a random graph. Proc. Natl. Acad. Sci. USA 107 4491–4498.
  • [10] Fountoulakis, N. and Reed, B. A. (2008). The evolution of the mixing rate of a simple random walk on the giant component of a random graph. Random Structures Algorithms 33 68–86.
  • [11] Leskovec, J., Lang, K. J., Dasgupta, A. and Mahoney, M. W. (2009). Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Math. 6 29–123.
  • [12] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [13] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York.
  • [14] Oliveira, R. I. (2012). On the coalescence time of reversible random walks. Trans. Amer. Math. Soc. 364 2109–2128.
  • [15] Pete, G. (2008). A note on percolation on $\mathbb{Z}^{d}$: Isoperimetric profile via exponential cluster repulsion. Electron. Commun. Probab. 13 377–392.
  • [16] Prata, A. (2012). Stochastic processes over finite graphs. Ph.D. thesis in Mathematics. IMPA, Rio de Janeiro, Brazil.