Random walks on generating sets for finite groups

Colloquium Mathematicum

From the author’s introduction: Let G be a finite group and let S be a set of generators of G. Suppose that S is not contained in a coset of a subgroup of G. Then for every probability measure μ such that suppμ=S we have lim n→∞ |μ *n -λ| X =0, where λ is the equidistributed probability measure on G: λ(g)=1/|G|, and |·| X denotes a suitable norm on the space of functions on G. The author is interested in questions concerning comparison of speeds of convergence to λ.

2005

Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitely solved, to get closed form formulas for the drift. The examples considered are the modular group Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy, and the volume of the group.

Proceedings of the American Mathematical Society

We describe the large time asymptotic behaviors of the probabilities p2t(e,e) of return to the origin associated to finite symmetric generating sets of abelian-by-cyclic groups. We characterize the different asymptotic behaviors by simple algebraic properties of the groups.

Annali di Matematica Pura ed Applicata, 1988

We consider the random walk (X~) associated with a probability p on a #ee produet o] discrete groups. Knowledge o] the resolvent (or Green's ]unction) o] p yields theorems about the asymptotic behaviour o] the n-step transition probabilities p*~(x) = P(X n = x I X o = e) as n-> c~. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour o] the type p*~(x)~ C~p-~n-~. Here we show on the other hand that i]G is a #so product o] ~ copies o] Z r and i] (Xn) is the ((average ~ o] the classical nearest neighbour random walk on each o] the ]actors Z% then while it satis]ies an (~ n-~-law ~) ]or r small relative to m, it switches to an n-r/2-1aw ]or large r. Using the same techniques, we give examples o] irreducible probabilities (o] in]inite support) on the #ee group Z *~ which satis]y n-~-laws ]or 2 #-~- .

Journal of Geometric Analysis, 2000

We show that, for random walks on Cayley graphs, the long time behavior of the probability of return after 2n steps is invariant by quasi-isometry. 1. Introduction Let G be a finitely generated group. For any finite generating set S satisfying S = S-1 , consider the Cayley graph (G, S) with vertex set G and an edge from x to y if and only if y = xs for some s ~ S. Thus, edges are oriented but this is merely a convention since (x, y) is an edge if and only if (y, x) is an edge. We allow the identity element id to be in S in which case our graph has a loop at each vertex. Clearly the graph (G, S) is invariant under the left action of G. Denote by Ixl the distance from the neutral element id to x in the Cayley graph (G, S), that is, ]xl is the minimal number k of elements of S needed to write x as x = sis2 .. 9 sk, si ~ S. The volume growth function of (G, S) is defined by V(n) =#Ix ~ a : Ixl _< n]. This paper focuses on the probability of return after 2n steps of the simple random walk on (G, S). For a survey of this topic, see [36]. The simple random walk on (G, S) is the Markov process (Xi)~ c with values in G which evolves as follows: If the current state is x, the next state is a neighbor of x chosen uniformly at random. This implicitly defines a probability measure Ps on G r~ such that Ps (Xn = y/ Xo = x) = Iz (sn' (x-ly) where 1 Ixs(g) = ~-~ls(g) and/1 ~n) is the n-fold convolution power of/x. Following usual notation we will also write P~(.) = Ps('/Xo = x) for the law of the walk based on S and started at x e G. To avoid parity problems, we consider only the probability of return at even times and set 4)s(n) = P~ (XEn = id) =/x(s2n)(id) 9

Journal of Theoretical Probability, 2002

We bound the rate of convergence to uniformity for a certain random walk on the complete monomial groups G≀S n for any group G. Specifically, we determine that \({\raise0.7ex\hbox{\(1\)} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em}\!\lower0.7ex\hbox{\(2\)}}\) n log n+ \(\frac{1}{4}\) n log (|G|−1|) steps are both necessary and sufficient for ℓ2 distance to become small. We also determine that \({\raise0.7ex\hbox{\(1\)} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em}\!\lower0.7ex\hbox{\(2\)}}\) n log n steps are both necessary and sufficient for total variation distance to become small. These results provide rates of convergence for random walks on a number of groups of interest: the hyperoctahedral group ℤ2≀S n , the generalized symmetric group ℤ m ≀S n , and S m ≀S n . In the special case of the hyperoctahedral group, our random walk exhibits the “cutoff phenomenon.”

2005

This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mairesse and F.Math\'eus. It contains the details of the computations and the proofs of the results concerning the examples treated there.

For every 3/4 &lt;= beta &lt; 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point is within a constant factor of n^beta. In fact, the speed can be set precisely to equal any nice prescribed function up to a constant factor.

Journal of the European Mathematical Society, 2003

We establish the lower bound p 2t (e, e) exp(−t 1/3), for the large times asymptotic behaviours of the probabilities p 2t (e, e) of return to the origin at even times 2t, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer r, such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to r.) Key words. random walk-heat kernel decay-asymptotic invariants of infinite groups-Prüfer rank-solvable group Contents

Nagoya Mathematical Journal, 1986

Suppose that G is a discrete group and p is a probability measure on G. Consider the associated random walk {Xn} on G. That is, let Xn = Y1Y2 … Yn, where the Yj’s are independent and identically distributed G-valued variables with density p. An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p. For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x) permits a description of the asymptotic behaviour of these probabilities.