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Abstract

We study the equidistribution on spheres of the n-step transition probabilities of random walks on graphs. We give sufficient conditions for this property being satisfied and for the weaker property of asymptotical equidistribution. We analyze the asymptotical behaviour of the Green function of the simple random walk on ℤ2 and we provide a class of random walks on Cayley graphs of groups, whose transition probabilities are not even asymptotically equidistributed.

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REFERENCES

  1. A. Avez, Limite des quotients pour des marches aláatoires sur des groupes, C. R. Acad. Sci. Paris Sár. A 276:317 (1973).

    Google Scholar 

  2. W. Woess, Nearest neighbour random walks on free product of discrete groups, Bollettino U.M.I. (6) 5–B:961 (1986).

    Google Scholar 

  3. D. I. Cartwright and P. M. Soardi, Random walks on free products, quotients and amalgams, Nagoya Math. J. 102:163–180 (1986).

    Google Scholar 

  4. D. I. Cartwright, Some examples of random walks on free products of discrete groups, Ann. Mat. Pura ed Appl. (IV) 151:1–15 (1988).

    Google Scholar 

  5. S. K. Nechaev, A. Yu. Grosberg, and A. M. Vershik, Random walks on braid groups: Brownian bridges, complexity and statistics, J. Phys. A 29:2411 (1996).

    Google Scholar 

  6. H. D. Macpherson, Infinite distance transitive graphs of finite valency, Combinatorica 2:63 (1982).

    Google Scholar 

  7. A. A. Ivanov, Bounding the diameter of a distance-regular graph, Soviet Math. Dokl. 28:149 (1983).

    Google Scholar 

  8. S. Sawyer, Isotropic random walks in a tree, Z. Wahrsch. Verw. Gebiete 42:279 (1978).

    Google Scholar 

  9. W. Woess, Random walks on infinite graphs and groups--A survey on selected topics, Bull. London Math. Soc. 26:1–60 (1994).

    Google Scholar 

  10. D. Cassi and S. Regina, Random walks on d-dimensional comb lattices, Modern Phys. Lett. B 6:1397 (1992).

    Google Scholar 

  11. M. Picardello and W. Woess, The full Martin boundary of the bi-tree, Annals of Probability 22:2203 (1994).

    Google Scholar 

  12. P. Ney and F. Spitzer, The Martin boundary for random walk, Trans. Amer. Math. Soc. 121:116 (1966).

    Google Scholar 

  13. J. Dixmier, Les moyennes invariantes dans les sámigroupes et leurs applications, Acta Sci. Math. (Szeged) 12A:213 (1950).

    Google Scholar 

  14. P. Gerl, A local central limit theorem on some groups, in The First Pannonian Symposium on Mathematical Statistics, Springer Lecture Notes in Statistics, Vol. 8, p. 73 (1981).

  15. M. Picardello and W. Woess, Random walks on amalgams, Monatsh. Math. 100:21 (1985).

    Google Scholar 

  16. Y. Guivarc'h, Sur la loi des grand nombres et le rayon spectral d'une marche aléatoire, Astérisque 74:47 (1980).

    Google Scholar 

  17. W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, in preparation.

Download references

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Bertacchi, D., Zucca, F. Equidistribution of Random Walks on Spheres. Journal of Statistical Physics 94, 91–111 (1999). https://doi.org/10.1023/A:1004559228814

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