Estimates for simple random walks on fundamental groups of surfaces

Laurent Bartholdi; Serge Cantat; Tullio Ceccherini-Silberstein; Pierre de la Harpe

Colloquium Mathematicae (1997)

  • Volume: 72, Issue: 1, page 173-193
  • ISSN: 0010-1354

Abstract

top
Numerical estimates are given for the spectral radius of simple random walks on Cayley graphs. Emphasis is on the case of the fundamental group of a closed surface, for the usual system of generators.

How to cite

top

Bartholdi, Laurent, et al. "Estimates for simple random walks on fundamental groups of surfaces." Colloquium Mathematicae 72.1 (1997): 173-193. <http://eudml.org/doc/210451>.

@article{Bartholdi1997,
abstract = {Numerical estimates are given for the spectral radius of simple random walks on Cayley graphs. Emphasis is on the case of the fundamental group of a closed surface, for the usual system of generators.},
author = {Bartholdi, Laurent, Cantat, Serge, Ceccherini-Silberstein, Tullio, de la Harpe, Pierre},
journal = {Colloquium Mathematicae},
keywords = {simple random walk; surface group; spectral radius; spectral radius of simple random walks on Cayley graphs; Poisson kernels; inequalities},
language = {eng},
number = {1},
pages = {173-193},
title = {Estimates for simple random walks on fundamental groups of surfaces},
url = {http://eudml.org/doc/210451},
volume = {72},
year = {1997},
}

TY - JOUR
AU - Bartholdi, Laurent
AU - Cantat, Serge
AU - Ceccherini-Silberstein, Tullio
AU - de la Harpe, Pierre
TI - Estimates for simple random walks on fundamental groups of surfaces
JO - Colloquium Mathematicae
PY - 1997
VL - 72
IS - 1
SP - 173
EP - 193
AB - Numerical estimates are given for the spectral radius of simple random walks on Cayley graphs. Emphasis is on the case of the fundamental group of a closed surface, for the usual system of generators.
LA - eng
KW - simple random walk; surface group; spectral radius; spectral radius of simple random walks on Cayley graphs; Poisson kernels; inequalities
UR - http://eudml.org/doc/210451
ER -

References

top
  1. [Can] J. W. Cannon, The growth of the closed surface groups and compact hyperbolic Coxeter groups, circulated typescript, University of Wisconsin, 1980. 
  2. [Car] D. I. Cartwright, Some examples of random walks on free products of discrete groups, Ann. Mat. Pura Appl. 151 (1988), 1-15. Zbl0661.60018
  3. [CaM] D. I. Cartwright and W. Młotkowski, Harmonic analysis for groups acting on triangle buildings, J. Austral. Math. Soc. Ser. A 56 (1994), 345-383. Zbl0808.51014
  4. [Cha] C. Champetier, Propriétés statistiques des groupes de présentation finie, Adv. in Math. 116 (1995), 197-262. 
  5. [ChM] B. Chandler and W. Magnus, The History of Combinatorial Group Theory: a Case Study in the History of Ideas, Springer, 1982. Zbl0498.20001
  6. [ChV] P. A. Cherix and A. Valette, On spectra of simple random walks on one-relator groups, Pacific J. Math., to appear. Zbl0865.60059
  7. [CdV] Y. Colin de Verdière, Spectres de graphes, prépublication, Grenoble, 1995. 
  8. [DoK] J. Dodziuk and L. Karp, Spectra and function theory for combinatorial Laplacians, in: Contemp. Math. 73, Amer. Math. Soc., 1988, 25-40. 
  9. [FP] W. J. Floyd and S. P. Plotnick, Symmetries of planar growth functions, Invent. Math. 93 (1988), 501-543. Zbl0652.20052
  10. [Har] T. E. Harris, Transient Markov chains with stationary measures, Proc. Amer. Math. Soc. 8 (1957), 937-942. Zbl0087.13501
  11. [Ke1] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336-354. Zbl0092.33503
  12. [Ke2] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146-156. Zbl0092.26704
  13. [LyS] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, 1977. 
  14. [Nag] T. Nagnibeda, An estimate from above of spectral radii of random walks on surface groups, Sbornik Seminarov POMI, A. Vershik (ed.), to appear. Zbl0947.60006
  15. [Pas] W. B. Paschke, Lower bound for the norm of a vertex-transitive graph, Math. Zeit. 213 (1993), 225-239. Zbl0798.05036
  16. [Pru] W. E. Pruitt, Eigenvalues of non-negative matrices, Ann. Math. Statist. 35 (1964), 1797-1800. Zbl0211.48502
  17. [Ser] C. Series, The infinite word problem and limit sets of Fuchsian groups, Ergodic Theory Dynam. Systems 1 (1981) 337-360. Zbl0483.30029
  18. [Sul] D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential. Geom. 25 (1987), 327-351. Zbl0615.53029
  19. [Wag] P. Wagreich, The growth function of a discrete group, in: Lecture Notes in Math. 956, Springer, 1982, 125-144. 
  20. [Woe] W. Woess, Random walks on infinite graphs and groups - a survey on selected topics, Bull. London Math. Soc. 26 (1994), 1-60. Zbl0830.60061
  21. [Żuk] A. Żuk, A remark on the norm of a random walk on surface groups, this volume, 195-206. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.