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Existence of graphs with sub exponential transitions probability decay and applications

Clément Rau

Bulletin de la Société Mathématique de France (2010)

  • Volume: 138, Issue: 4, page 491-542
  • ISSN: 0037-9484

Abstract

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In this paper, we recall the existence of graphs with bounded valency such that the simple random walk has a return probability at time n at the origin of order exp ( - n α ) , for fixed α ∈ [ 0 , 1 [ and with Følner function exp ( n 2 α 1 - α ) . This result was proved by Erschler (see [4], [3]); we give a more detailed proof of this construction in the appendix. In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random walk on an infinite cluster on the percolation model.

How to cite

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Rau, Clément. "Existence of graphs with sub exponential transitions probability decay and applications." Bulletin de la Société Mathématique de France 138.4 (2010): 491-542. <http://eudml.org/doc/272448>.

@article{Rau2010,
abstract = {In this paper, we recall the existence of graphs with bounded valency such that the simple random walk has a return probability at time $n$ at the origin of order $\exp (-n^\{\alpha \}),$ for fixed $\alpha \in [0,1[$ and with Følner function $\exp (n^\{ \frac\{2\alpha \}\{1-\alpha \}\})$. This result was proved by Erschler (see [4], [3]); we give a more detailed proof of this construction in the appendix. In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random walk on an infinite cluster on the percolation model.},
author = {Rau, Clément},
journal = {Bulletin de la Société Mathématique de France},
keywords = {random walk; local time; percolation cluster; isoperimetric inequality; wreath product; generalized wreath product},
language = {eng},
number = {4},
pages = {491-542},
publisher = {Société mathématique de France},
title = {Existence of graphs with sub exponential transitions probability decay and applications},
url = {http://eudml.org/doc/272448},
volume = {138},
year = {2010},
}

TY - JOUR
AU - Rau, Clément
TI - Existence of graphs with sub exponential transitions probability decay and applications
JO - Bulletin de la Société Mathématique de France
PY - 2010
PB - Société mathématique de France
VL - 138
IS - 4
SP - 491
EP - 542
AB - In this paper, we recall the existence of graphs with bounded valency such that the simple random walk has a return probability at time $n$ at the origin of order $\exp (-n^{\alpha }),$ for fixed $\alpha \in [0,1[$ and with Følner function $\exp (n^{ \frac{2\alpha }{1-\alpha }})$. This result was proved by Erschler (see [4], [3]); we give a more detailed proof of this construction in the appendix. In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random walk on an infinite cluster on the percolation model.
LA - eng
KW - random walk; local time; percolation cluster; isoperimetric inequality; wreath product; generalized wreath product
UR - http://eudml.org/doc/272448
ER -

References

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  1. [1] T. Coulhon – « Ultracontractivity and Nash type inequalities », J. Funct. Anal.141 (1996), p. 510–539. Zbl0887.58009MR1418518
  2. [2] A. Erschler – « On isoperimetric profiles of finitely generated groups », Geom. Dedicata100 (2003), p. 157–171. Zbl1049.20024MR2011120
  3. [3] —, « Generalized wreath products », Int. Math. Res. Not. (2006), p. 14. Zbl1118.05042MR2276348
  4. [4] —, « Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks », Probab. Theory Related Fields136 (2006), p. 560–586. Zbl1105.60009MR2257136
  5. [5] M. Gromov – « Entropy and isoperimetry for linear and non-linear group actions », preprint http://www.ihes.fr/~gromov/topics/grig-final-june11-08.pdf. Zbl1280.20043MR2442946
  6. [6] P. Mathieu & E. Remy – « Isoperimetry and heat kernel decay on percolation clusters », Ann. Probab.32 (2004), p. 100–128. Zbl1078.60085MR2040777
  7. [7] C. Rau – « Marches aléatoires sur un amas infini de percolation », Thèse, Université de Provence - Aix-Marseille I, 2006. 
  8. [8] —, « Sur le nombre de points visités par une marche aléatoire sur un amas infini de percolation », Bull. Soc. Math. France135 (2007), p. 135–169. Zbl1156.60074MR2430203
  9. [9] L. Saloff-Coste & C. Pittet – « A survey on the relationships between volume growth, isoperimetry, and the behaviour of simple random walks on Caley graphs, with example », preprint, 2001. 
  10. [10] W. Woess – Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, 2000. Zbl0951.60002MR1743100

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