The rate of escape for random walks on polycyclic and metabelian groups

Russ Thompson

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 1, page 270-287
  • ISSN: 0246-0203

Abstract

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We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on some abelian-by-cyclic groups via a comparison to the toppling of a dissipative abelian sandpile.

How to cite

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Thompson, Russ. "The rate of escape for random walks on polycyclic and metabelian groups." Annales de l'I.H.P. Probabilités et statistiques 49.1 (2013): 270-287. <http://eudml.org/doc/271951>.

@article{Thompson2013,
abstract = {We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on some abelian-by-cyclic groups via a comparison to the toppling of a dissipative abelian sandpile.},
author = {Thompson, Russ},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {law of iterated logarithm; metabelian group; polycyclic group; random walk; rate of escape; abelian sandpile; solvable group; subgroup distortion},
language = {eng},
number = {1},
pages = {270-287},
publisher = {Gauthier-Villars},
title = {The rate of escape for random walks on polycyclic and metabelian groups},
url = {http://eudml.org/doc/271951},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Thompson, Russ
TI - The rate of escape for random walks on polycyclic and metabelian groups
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 1
SP - 270
EP - 287
AB - We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on some abelian-by-cyclic groups via a comparison to the toppling of a dissipative abelian sandpile.
LA - eng
KW - law of iterated logarithm; metabelian group; polycyclic group; random walk; rate of escape; abelian sandpile; solvable group; subgroup distortion
UR - http://eudml.org/doc/271951
ER -

References

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  1. [1] T. Austin, A. Naor and Y. Peres. The wreath product of ℤ with ℤ has Hilbert compression exponent 2 3 . Proc. Amer. Math. Soc.137 (2009) 85–90. Zbl1226.20032MR2439428
  2. [2] M. T. Barlow and E. A. Perkins. Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields79 (1988) 543–623. Zbl0635.60090MR966175
  3. [3] G. Baumslag. Subgroups of finitely presented metabelian groups. J. Aust. Math. Soc. 16 (1973) 98–110. Collection of articles dedicated to the memory of Hanna Neumann, I. Zbl0287.20027MR332999
  4. [4] I. Benjamini and D. Revelle. Instability of set recurrence and Green’s function on groups with the Liouville property. Potential Anal.34 (2011) 199–206. Zbl1223.05286MR2754971
  5. [5] G. R. Conner. Discreteness properties of translation numbers in solvable groups. J. Group Theory3 (2000) 77–94. Zbl0956.20039MR1736519
  6. [6] T. Davis and A. Olshanskii. Subgroup distortion in wreath products of cyclic groups. J. Pure Appl. Algebra215 (2011) 2987–3004. Zbl1259.20049MR2811580
  7. [7] Y. Derriennic. Quelques applications du théorème ergodique sous-additif. In Conference on Random Walks (Kleebach, 1979) 183–201. Astérisque 74. Soc. Math. France, Paris, 1980. Zbl0446.60059MR588163
  8. [8] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge, 2010. Zbl1202.60001MR2722836
  9. [9] A. Erschler. On drift and entropy growth for random walks on groups. Ann. Probab.31 (2003) 1193–1204. Zbl1043.60005MR1988468
  10. [10] A. Erschler. Critical constants for recurrence of random walks on G -spaces. Ann. Inst. Fourier (Grenoble) 55 (2005) 493–509. Zbl1133.20031MR2147898
  11. [11] S. M. Gersten. Preservation and distortion of area in finitely presented groups. Geom. Funct. Anal.6 (1996) 301–345. Zbl0868.20033MR1384614
  12. [12] A. Grigor’yan. Escape rate of Brownian motion on Riemannian manifolds. Appl. Anal.71 (1999) 63–89. Zbl1020.58024MR1690091
  13. [13] Y. Guivarc’h. Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. In Conference on Random Walks (Kleebach, 1979) 47–98. Astérisque 74. Soc. Math. France, Paris, 1980. Zbl0448.60007MR588157
  14. [14] W. Hebisch and L. Saloff-Coste. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab.21 (1993) 673–709. Zbl0776.60086MR1217561
  15. [15] V. A. Kaĭmanovich and A. M. Vershik. Random walks on discrete groups: Boundary and entropy. Ann. Probab.11 (1983) 457–490. Zbl0641.60009MR704539
  16. [16] J. F. C. Kingman. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B30 (1968) 499–510. Zbl0182.22802MR254907
  17. [17] J. R. Lee and Y. Peres. Harmonic maps on amenable groups and a diffusive lower bound for random walks. Preprint, 2009. Zbl1284.05250MR3127886
  18. [18] V. Nekrashevych. Self-Similar Groups. Mathematical Surveys and Monographs 117. Amer. Math. Soc., Providence, RI, 2005. Zbl1087.20032MR2162164
  19. [19] D. V. Osin. Exponential radicals of solvable Lie groups. J. Algebra248 (2002) 790–805. Zbl1001.22006MR1882124
  20. [20] C. Pittet. Følner sequences in polycyclic groups. Rev. Mat. Iberoamericana11 (1995) 675–685. Zbl0842.20035MR1363210
  21. [21] F. Redig. Mathematical aspects of the abelian sandpile model. In Mathematical Statistical Physics 657–729. Elsevier, Amsterdam, 2006. Zbl05723807MR2581895
  22. [22] D. Revelle. Rate of escape of random walks on wreath products and related groups. Ann. Probab.31 (2003) 1917–1934. Zbl1051.60047MR2016605
  23. [23] P. Révész. Random Walk in Random and Non-Random Environments, 2nd edition. World Scientific, Hackensack, NJ, 2005. Zbl1283.60007
  24. [24] D. Segal. Polycyclic Groups. Cambridge Tracts in Mathematics 82. Cambridge Univ. Press, Cambridge, 1983. Zbl0516.20001MR713786
  25. [25] R. Tessera. Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces. Comment. Math. Helv.86 (2011) 499–535. Zbl1274.43009MR2803851
  26. [26] E. Teufl. The average displacement of the simple random walk on the Sierpiński graph. Combin. Probab. Comput.12 (2003) 203–222. Zbl1025.60034MR1967404
  27. [27] N. T. Varopoulos. Long range estimates for Markov chains. Bull. Sci. Math.109 (1985) 225–252. Zbl0583.60063MR822826
  28. [28] A. D. Warshall. Deep pockets in lattices and other groups. Trans. Amer. Math. Soc.362 (2010) 577–601. Zbl1269.20034MR2551498

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