Annals of Applied Probability

On meteors, earthworms and WIMPs

Sara Billey, Krzysztof Burdzy, Soumik Pal, and Bruce E. Sagan

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We study a model of mass redistribution on a finite graph. We address the questions of convergence to equilibrium and the rate of convergence. We present theorems on the distribution of empty sites and the distribution of mass at a fixed vertex. These distributions are related to random permutations with certain peak sets.

Article information

Ann. Appl. Probab., Volume 25, Number 4 (2015), 1729-1779.

Received: August 2013
Revised: March 2014
First available in Project Euclid: 21 May 2015

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Meteor process mass redistribution Markov processes on graphs permutation statistics


Billey, Sara; Burdzy, Krzysztof; Pal, Soumik; Sagan, Bruce E. On meteors, earthworms and WIMPs. Ann. Appl. Probab. 25 (2015), no. 4, 1729--1779. doi:10.1214/14-AAP1035.

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  • [1] Aldous, D. and Fill, J. (2014). Reversible Markov Chains and Random Walks on Graphs. Book in preparation. Available at
  • [2] Aldous, D. J. (1991). Meeting times for independent Markov chains. Stochastic Process. Appl. 38 185–193.
  • [3] Billey, S., Burdzy, K. and Sagan, B. E. (2013). Permutations with given peak set. J. Integer Seq. 16 Article 13.6.1, 18.
  • [4] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [5] Bóna, M. (2007). The copies of any permutation pattern are asymptotically normal. Available at arXiv:0712.2792.
  • [6] Burdzy, K. (2013). Meteor process on $\mathbb{Z}^{d}$. Available at arXiv:1312.6865.
  • [7] Burdzy, K., Chen, Z.-Q. and Pal, S. (2013). Brownian earthworm. Ann. Probab. 41 4002–4049.
  • [8] Caputo, P., Liggett, T. M. and Richthammer, T. (2010). Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc. 23 831–851.
  • [9] Chan, O.-Y. and Prałat, P. (2012). Chipping away at the edges: How long does it take? J. Comb. 3 101–121.
  • [10] Chao, C.-C. (1997). A note on applications of the martingale central limit theorem to random permutations. Random Structures Algorithms 10 323–332.
  • [11] Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg.
  • [12] Conger, M. and Viswanath, D. (2007). Normal approximations for descents and inversions of permutations of multisets. J. Theoret. Probab. 20 309–325.
  • [13] Crane, H. (2014). The cut-and-paste process. Ann. Probab. 42 1952–1979.
  • [14] Crane, H. and Lalley, S. P. (2013). Convergence rates of Markov chains on spaces of partitions. Electron. J. Probab. 18 1–23.
  • [15] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  • [16] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41 45–76.
  • [17] Dyson, F. J. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191–1198.
  • [18] Ehrenborg, R. and Mahajan, S. (1998). Maximizing the descent statistic. Ann. Comb. 2 111–129.
  • [19] Ferrari, P. A. and Fontes, L. R. G. (1998). Fluctuations of a surface submitted to a random average process. Electron. J. Probab. 3 34 pp. (electronic).
  • [20] Fey-den Boer, A., Meester, R., Quant, C. and Redig, F. (2008). A probabilistic approach to Zhang’s sandpile model. Comm. Math. Phys. 280 351–388.
  • [21] Furstenberg, H. and Kesten, H. (1960). Products of random matrices. Ann. Math. Statist. 31 457–469.
  • [22] Hairer, M., Mattingly, J. C. and Scheutzow, M. (2011). Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Related Fields 149 223–259.
  • [23] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series 51. Amer. Math. Soc., Providence, RI.
  • [24] Howitt, C. and Warren, J. (2009). Consistent families of Brownian motions and stochastic flows of kernels. Ann. Probab. 37 1237–1272.
  • [25] Kasraoui, A. (2012). The most frequent peak set of a random permutation. Available at arXiv:1210.5869.
  • [26] Kermack, W. O. and McKendrick, A. G. (1937). Some distributions associated with a randomly arranged set of numbers. Proc. Roy. Soc. Edinburgh 57 332–376.
  • [27] Kermack, W. O. and McKendrick, A. G. (1937). Tests for randomness in a series of numerical observations. Proc. Roy. Soc. Edinburgh 57 228–240.
  • [28] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [29] Wolfram Research (2010). Mathematica. Version 8.0. Wolfram Research, Champaign, IL.