## Electronic Journal of Probability

### Random Walks on Groups and Monoids with a Markovian Harmonic Measure

Mairesse Jean

#### Abstract

We consider a transient nearest neighbor random walk on a group $G$ with finite set of generators $S$. The pair $(G,S)$ is assumed to admit a natural notion of normal form words where only the last letter is modified by multiplication by a generator. The basic examples are the free products of a finitely generated free group and a finite family of finite groups, with natural generators. We prove that the harmonic measure is Markovian of a particular type. The transition matrix is entirely determined by the initial distribution which is itself the unique solution of a finite set of polynomial equations of degree two. This enables to efficiently compute the drift, the entropy, the probability of ever hitting an element, and the minimal positive harmonic functions of the walk. The results extend to monoids.

#### Article information

Source
Electron. J. Probab., Volume 10 (2005), paper no. 43, 1417-1441.

Dates
Accepted: 16 December 2005
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464816844

Digital Object Identifier
doi:10.1214/EJP.v10-293

Mathematical Reviews number (MathSciNet)
MR2191634

Zentralblatt MATH identifier
1109.60037

Rights

#### Citation

Jean, Mairesse. Random Walks on Groups and Monoids with a Markovian Harmonic Measure. Electron. J. Probab. 10 (2005), paper no. 43, 1417--1441. doi:10.1214/EJP.v10-293. https://projecteuclid.org/euclid.ejp/1464816844

#### References

• C. Aliprantis and K. Border. Infinite Dimensional Analysis: a Hitchhiker's Guide, 2nd ed. (1999) Springer-Verlag.
• A. Avez. Entropie des groupes de type fini. C. R. Acad. Sci. Paris Sér. A-B 275 (1972), 1363–1366.
• D. Cartwright and B. Krön. Classification of 0-automatic pairs. Preprint (2004).
• Y. Derriennic. Quelques applications du théorème ergodique sous-additif. Astérisque 74 (1980), 183-201.
• Y. Derriennic. Entropie, théorèmes limite et marches aléatoires. Probability measures on groups, VIII (Oberwolfach, 1985) Lecture Notes in Math. 1210 (1986), 241-284, Springer.
• E. Dynkin and M. Malyutov. Random walk on groups with a finite number of generators. Sov. Math. Dokl. 2 (1961), 399–402.
• D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, and W. Thurston. Word processing in groups. (1992) Jones and Bartlett.
• H. Furstenberg. Noncommuting random products. Trans. Amer. Math. Soc. 108 (1963), 377-428.
• H. Furstenberg. Random walks and discrete subgroups of Lie groups. Advances Probab. Related Topics, Vol. 1 (1971), 1-63, Dekker.
• M. Gromov. Hyperbolic groups. Essays in group theory. Math. Sci. Res. Inst. Publ. 8 (1987), 75-263, Springer.
• Y. Guivarc'h. Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire. Astérisque 74 (1980), 47-98.
• R. Haring-Smith. Groups and simple languages. Trans. Amer. Math. Soc. 279 (1983), no. 1, 337-356.
• G. Högnäs and A. Mukherjea. Probability measures on semigroups : convolution products, random walks, and random matrices. The University Series in Mathematics (1995), Plenum Press.
• V. Kaimanovich. The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2) 152 (2000), no. 3, 659-692.
• V. Kaimanovich and A. Vershik. Random walks on discrete groups: boundary and entropy. Ann. Probab. 11 (1983), no. 3, 457-490.
• J.F.C. Kingman. Subadditive ergodic theory. Ann. Probab. 1 (1973), 883-909.
• S. Lalley. Finite range random walk on free groups and homogeneous trees. Ann. Probab. 21 (1993), no. 4, 2087-2130.
• S. Lalley. Random walks on regular languages and algebraic systems of generating functions. Algebraic methods in statistics and probability. Contemp. Math. 287 (2001), 201-230, Amer. Math. Soc.
• F. Ledrappier. Some asymptotic properties of random walks on free groups. Topics in probability and Lie groups: boundary theory. CRM Proc. Lect. Notes 28 (2001), 117-152, Amer. Math. Soc.
• R. Lyons. Random walks and the growth of groups. C. R. Acad. Sci., Paris, Sér. I 320 (1995), no. 11, 1361-1366.
• J. Mairesse. Zero-automaticity for groups and monoids. Preprint available at http://www.liafa.jussieu.fr/~mairesse/Article (2004).
• J. Mairesse and F. Mathéus. Random walks on free products of cyclic groups. arXiv:math.PR/0509211 (2005).
• J. Mairesse and F. Mathéus. Random walks on groups with a tree-like Cayley graph. Mathematics and computer science. III. Algorithms, trees, combinatorics and probabilities. Trends Math. (2004), 445-460, Birkhäuser.
• T. Nagnibeda and W. Woess. Random walks on trees with finitely many cone types. J. Theoret. Probab. 15 (2002), no. 2, 383-422.
• S. Sawyer and T. Steger. The rate of escape for anisotropic random walks in a tree. Probab. Theory Related Fields 76 (1987), no. 2, 207-230.
• P. Soardi. Limit theorems for random walks on discrete semigroups related to nonhomogeneous trees and Chebyshev polynomials. Math. Z. 200 (1989), no. 3, 313-325.
• J. Stallings. A remark about the description of free products of groups. Proc. Cambridge Philos. Soc. 62 (1966), 129-134.
• C. Takacs. Random walk on periodic trees. Electron. J. Probab. 2 (1997), no. 1, 1-16.
• A. Vershik. Dynamic theory of growth in groups: Entropy, boundaries, examples. Russ. Math. Surv. 55 (2000), no. 4, 667-733.
• W. Woess. A description of the Martin boundary for nearest neighbour random walks on free products. Probability measures on groups, VIII (Oberwolfach, 1985) Lecture Notes in Math. 1210 (1986), 203-215, Springer.
• W. Woess. Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics 138 (2000), Cambridge University Press.