The rate of escape for random walks on polycyclic and metabelian groups
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 1, pp. 270-287.

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.

Nous utilisons la notion de distorsion des sous-groupes afin de déterminer la vitesse de fuite (sous linéaire) d'une marche aléatoire simple sur une classe de groupes polycycliques, et nous montrons que cette vitesse est invariante par changement de générateurs pour ces groupes. Pour les groupes métabéliens, nous définissons une forme plus forte de distorsion des sous-groupes qui s'applique à des sous-groupes non finiment engendrés. Sous cette hypothèse, nous calculons la vitesse de fuite pour certaines marches aléatoires sur certains groupes abélien par cyclique via l'intermédiaire d'une comparaison avec la chute d'un tas de sable abélien dissipatif.

DOI: 10.1214/11-AIHP455
Classification: 60B15, 20F65
Keywords: law of iterated logarithm, metabelian group, polycyclic group, random walk, rate of escape, abelian sandpile, solvable group, subgroup distortion
@article{AIHPB_2013__49_1_270_0,
     author = {Thompson, Russ},
     title = {The rate of escape for random walks on polycyclic and metabelian groups},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {270--287},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {1},
     year = {2013},
     doi = {10.1214/11-AIHP455},
     mrnumber = {3060157},
     zbl = {1274.60018},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/11-AIHP455/}
}
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
SP  - 270
EP  - 287
VL  - 49
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/11-AIHP455/
DO  - 10.1214/11-AIHP455
LA  - en
ID  - AIHPB_2013__49_1_270_0
ER  - 
%0 Journal Article
%A Thompson, Russ
%T The rate of escape for random walks on polycyclic and metabelian groups
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2013
%P 270-287
%V 49
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/11-AIHP455/
%R 10.1214/11-AIHP455
%G en
%F AIHPB_2013__49_1_270_0
Thompson, Russ. The rate of escape for random walks on polycyclic and metabelian groups. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 1, pp. 270-287. doi : 10.1214/11-AIHP455. http://archive.numdam.org/articles/10.1214/11-AIHP455/

[1] T. Austin, A. Naor and Y. Peres. The wreath product of with has Hilbert compression exponent 2 3. Proc. Amer. Math. Soc. 137 (2009) 85-90. | MR | Zbl

[2] M. T. Barlow and E. A. Perkins. Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields 79 (1988) 543-623. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[5] G. R. Conner. Discreteness properties of translation numbers in solvable groups. J. Group Theory 3 (2000) 77-94. | MR | Zbl

[6] T. Davis and A. Olshanskii. Subgroup distortion in wreath products of cyclic groups. J. Pure Appl. Algebra 215 (2011) 2987-3004. | MR | Zbl

[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. | Numdam | MR | Zbl

[8] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge, 2010. | MR | Zbl

[9] A. Erschler. On drift and entropy growth for random walks on groups. Ann. Probab. 31 (2003) 1193-1204. | MR | Zbl

[10] A. Erschler. Critical constants for recurrence of random walks on G-spaces. Ann. Inst. Fourier (Grenoble) 55 (2005) 493-509. | Numdam | MR | Zbl

[11] S. M. Gersten. Preservation and distortion of area in finitely presented groups. Geom. Funct. Anal. 6 (1996) 301-345. | MR | Zbl

[12] A. Grigor'Yan. Escape rate of Brownian motion on Riemannian manifolds. Appl. Anal. 71 (1999) 63-89. | MR | Zbl

[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. | Numdam | MR | Zbl

[14] W. Hebisch and L. Saloff-Coste. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 (1993) 673-709. | MR | Zbl

[15] V. A. Kaĭmanovich and A. M. Vershik. Random walks on discrete groups: Boundary and entropy. Ann. Probab. 11 (1983) 457-490. | MR | Zbl

[16] J. F. C. Kingman. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30 (1968) 499-510. | MR | Zbl

[17] J. R. Lee and Y. Peres. Harmonic maps on amenable groups and a diffusive lower bound for random walks. Preprint, 2009. | MR

[18] V. Nekrashevych. Self-Similar Groups. Mathematical Surveys and Monographs 117. Amer. Math. Soc., Providence, RI, 2005. | MR | Zbl

[19] D. V. Osin. Exponential radicals of solvable Lie groups. J. Algebra 248 (2002) 790-805. | MR | Zbl

[20] C. Pittet. Følner sequences in polycyclic groups. Rev. Mat. Iberoamericana 11 (1995) 675-685. | MR | Zbl

[21] F. Redig. Mathematical aspects of the abelian sandpile model. In Mathematical Statistical Physics 657-729. Elsevier, Amsterdam, 2006. | MR

[22] D. Revelle. Rate of escape of random walks on wreath products and related groups. Ann. Probab. 31 (2003) 1917-1934. | MR | Zbl

[23] P. Révész. Random Walk in Random and Non-Random Environments, 2nd edition. World Scientific, Hackensack, NJ, 2005. | Zbl

[24] D. Segal. Polycyclic Groups. Cambridge Tracts in Mathematics 82. Cambridge Univ. Press, Cambridge, 1983. | MR | Zbl

[25] R. Tessera. Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces. Comment. Math. Helv. 86 (2011) 499-535. | MR | Zbl

[26] E. Teufl. The average displacement of the simple random walk on the Sierpiński graph. Combin. Probab. Comput. 12 (2003) 203-222. | MR | Zbl

[27] N. T. Varopoulos. Long range estimates for Markov chains. Bull. Sci. Math. 109 (1985) 225-252. | MR | Zbl

[28] A. D. Warshall. Deep pockets in lattices and other groups. Trans. Amer. Math. Soc. 362 (2010) 577-601. | MR | Zbl

Cited by Sources: