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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 90 open journal systems 


Recurrence and transience for long-range reversible random walks on a random point process

Pietro Caputo, Universitą di Roma Tre
Alessandra Faggionato, University La Sapienza, Rome. Italy
Alexandre Gaudilliere, University of Roma Tre.


Abstract
We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.


Full text: PDF

Pages: 2580­-2616

Published on: November 30, 2009
Bibliography
  1. L. Addario-Berry, A. Sarkar. The simple random walk on a random Voronoi tiling. Preprint.
  2. M.T. Barlow, R.F. Bass, T. Kumagai. Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps. Math. Z. 261 (2009), 297-320. Math. Review 2009m:60111
  3. N. Berger. Transience, recurrence and critical behavior for long-range percolation. Commun. Math. Phys. 226 (2002), 531-558. Math. Review 2003a:82034
  4. J. Ben Hough, M. Krishnapur, Y. Peres, B.Virág. Determinantal processes and independence. Probability Surveys. 3 (2006), 206-229. Math. Review 2006m:60068
  5. I. Benjamini, R. Pemantle, Y. Peres. Unpredictable paths and percolation. Ann. Probab. 26 (1998), 1198-1211. Math. Review 99g:60183
  6. P. Caputo, A. Faggionato. Isoperimetric inequalities and mixing time for a random walk on a random point process. Ann. Appl. Probab. 17 (2007), 1707-1744. Math. Review 2008m:60204
  7. P. Caputo, A. Faggionato. Diffusivity in one-dimensional generalized Mott variable-range hopping models. Ann. Appl. Probab. 19 (2009), 1459-1494. Math. Review number not available.
  8. D.J. Delay, D. Vere-Jones. An introduction to the theory of point processes. Vol. I, Second edition. Springer-Verlag, 2003.
  9. P.G. Doyle, J.L. Snell. Random walks and electric networks. The Carus mathematical monographs 22, Mathematical Association of America, Washington, 1984. Math. Review 89a:94023
  10. R.A. Doney. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Rel. Fields 107 (1997), 451-465. Math. Review 98e:60040
  11. A. Faggionato, P. Mathieu. Mott law as upper bound for a random walk in a random environment. Comm. Math. Phys. 281 (2008), 263-286. Math. Review 2009b:60309
  12. A. Faggionato, H. Schulz--Baldes, D. Spehner. Mott law as lower bound for a random walk in a random environment. Comm. Math. Phys. 263 (2006), 21-64. Math. Review 2007c:82034
  13. A. Gaudillière. Condenser physics applied to Markov chains - A brief introduction to potential theory. Lecture notes.
  14. H.-O. Georgii, T. Küneth. Stochastic comparison of point random fields. J. Appl. Probab. 34 (1997), 868-881. Math. Review 99c:60101
  15. G. Grimmett. Percolation. Second edition. Springer, Grundlehren 321, Berlin, 1999. Math. Review 2001a:60114
  16. G. Grimmett, H. Kesten, Y. Zhang. Random walk on the infinite cluster of the percolation model. Probab. Theory Rel. Fields 96, no. 1 (1993), 33-44. Math. Review 94i:60078
  17. T. Kumagai, J. Misumi. Heat kernel estimates for strongly recurrent random walk on random media. J. Theoret. Probab. 21 (2008), 910-935. Math. Review 2009g:35101
  18. J. Kurkijärvi. Hopping conductivity in one dimension. Phys. Rev. B, 8, no. 2 (1973), 922--924.
  19. T. Lyons. A simple criterion for transience of a reversible Markov chain. Ann. Probab. 11, no. 2 (1983), 393-402. Math. Review 84e:60102
  20. R. Lyons, Y. Peres. Probability on Trees and Networks. Book in progress.
  21. R. Lyons, J. Steif. Stationary determinantal processes: Phase multiplicity, Bernoullicity, Entropy, and Domination. Duke Math. Journal 120 (2003), 515-575. Math. Review 2004k:60100
  22. J. Misumi. Estimates on the effective resistance in a long-range percolation on Z^d. Kyoto U. Math. Journal 48, no.2 (2008), 389-400. Math. Review 2009d:60332
  23. A. Pisztora. Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Rel. Fields 104 (1996), 427-466. Math. Review 97d:82016
  24. A. Soshnikov. Determinantal random point fields. Russian Mathematical Surveys 55 (2000), 923-975. Math. Review 2002f:60097
  25. F. Spitzer. Principles of random walk . Second edition, Graduate Texts in Mathematics, Vol. 34. Springer-Verlag, 1976. Math. Review 52#9383
















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Electronic Journal of Probability. ISSN: 1083-6489