Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 15(2010) > Paper 23 open journal systems 


Critical constants for recurrence on groups of polynomial growth

David Revelle,
Russ M Thompson, Cornell University


Abstract
The critical constant for recurrence, crt, is an invariant of the quotient space H G of a finitely generated group. The constant is determined by the largest moment a probability measure on G can have without the induced random walk on H G being recurrent. We present a description of which subgroups of groups of polynomial volume growth are recurrent. Using this we show that for such recurrent subgroups crt corresponds to the relative growth rate of H in G, and in particular crt is either 0, 1 or 2.


Full text: PDF

Pages: 710-722

Published on: May 20, 2010
Bibliography
  1. Aldous, David J.; Fill, James Allen. Reversible MArkov Chains. Reversible Markov Chains and Random Walks on Graphs, 2002. http://www.stat.berkeley.edu/~aldous/RWG/book.html
  2. Bass, H. The degree of polynomial growth of finitely generated nilpotent groups. Proc. London Math. Soc. (3) 25 (1972), 603--614. MR0379672 (52 #577)
  3. de la Harpe, Pierre. Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. vi+310 pp. ISBN: 0-226-31719-6; 0-226-31721-8 MR1786869 (2001i:20081)
  4. Dynkin, Evgenii B.; Yushkevich, Aleksandr A. Markov processes: Theorems and problems. Translated from the Russian by James S. Wood Plenum Press, New York 1969 x+237 pp. MR0242252 (39 #3585a)
  5. Erschler, Anna. Boundary behavior for groups of subexponential growth. Ann. of Math. (2) 160 (2004), no. 3, 1183--1210. MR2144977 (2006d:20072)
  6. Erschler, Anna. Critical constants for recurrence of random walks on $G$-spaces. Ann. Inst. Fourier (Grenoble) 55 (2005), no. 2, 493--509. MR2147898 (2006c:20085)
  7. Gromov, Mikhael. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53 (1981), 53--73. MR0623534 (83b:53041)
  8. Gallardo, Léonard; Schott, René. Marches aléatoires sur les espaces homogènes de certains groupes de type rigide. (French) Conference on Random Walks (Kleebach, 1979) (French), pp. 149--170, 4, Astérisque, 74, Soc. Math. France, Paris, 1980. MR0588161 (82e:60107a)
  9. Guivarc'h, Yves. Groupes de Lie à croissance polynomiale. (French) C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1695--A1696. MR0302819 (46 #1962)
  10. Hebisch, W.; Saloff-Coste, L. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 (1993), no. 2, 673--709. MR1217561 (94m:60144)
  11. Lubotzky, Alexander. Cayley graphs: eigenvalues, expanders and random walks. Surveys in combinatorics, 1995 (Stirling), 155--189, London Math. Soc. Lecture Note Ser., 218, Cambridge Univ. Press, Cambridge, 1995. MR1358635 (96k:05081)
  12. Nekrashevych, Volodymyr. Self-similar groups. Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. xii+231 pp. ISBN: 0-8218-3831-8 MR2162164 (2006e:20047)
  13. Pólya, Georg. Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz. (German) Math. Ann. 84 (1921), no. 1-2, 149--160. MR1512028
  14. Revelle, David. Random walks on solvable groups. Thesis. Cornell University, 2002.
  15. Spitzer, Frank. Principles of random walks. Second edition. Graduate Texts in Mathematics, Vol. 34. Springer-Verlag, New York-Heidelberg, 1976. xiii+408 pp. MR0388547 (52 #9383)
  16. Varopoulos, Nicholas Th. Théorie du potentiel sur les groupes nilpotents. (French) [Potential theory on nilpotent groups] C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 5, 143--144. MR0801947 (86i:22017)
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489