Some examples of random walks on free products of discrete groups (original) (raw)

Random walks on free products, quotients and amalgams

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.

On the stability of the behavior of random walks on groups

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

Random walks on free products of cyclic groups

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.

Some remarks on the random walk on 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.

Random Walks in I.I.D. Random Environment on Non-Abelian Free Groups

We consider the random walk on an independent and identically distributed (i.i.d.) random environment on a Cayley graph of a finitely generated non-abelian free group. Such a Cayley graph is readily seen to be a regular tree with even degree. Under a non-degeneracy assumption we show that the walk is always transient.

Random walks on generating sets for finite groups

The Electronic Journal of Combinatorics - Electr. J. Comb., 1997

We analyze a certain random walk on the cartesian product Gn of a flnite group G which is often used for generating random elements from G. In particular, we show that the mixing time of the walk is at most crn2 logn where the constant cr depends only on the order r of G.

Random walks on finite rank solvable groups

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

Random Walks on Abelian-by-Cyclic Groups

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.

Reciprocal class of random walks on an Abelian group

2015

Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.