Suppose we are given the free product V of a finite family of finite or countable sets (V/), e jj and probability measures on each V h which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the Vj. We prove the existence of the rate of escape with respect to the block length, that is, the speed at which the random walk escapes to infinity, and furthermore we compute formulae for it. For this purpose, we present three different techniques providing three different, equivalent formulae.2000 Mathematics subject classification: primary 60G50; secondary 20E06, 60B15.

Abstract. Suppose we are given finitely generated groups Γ1, . . . , Γm equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product Γ1 * . . . * Γm and give a complete classification of the possible asymptotic behaviour of the corresponding n-step return probabilities. They either inherit a law of the form ̺ nδ n −λ i log κ i n from one of the free factors Γi or obey a ̺ nδ n −3/2 -law, where ̺ < 1 is the corresponding spectral radius and δ is the period of the random walk. In addition, we determine the full range of the asymptotic behaviour in the case of nearest neighbour random walks on free products of the form Z d 1 * . . . * Z dm . Moreover, we characterize the possible phase transitions of the non-exponential types n −λ i log κ i n in the case Γ1 * Γ2.

Abstract. Suppose we are given the free product V of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors. We prove the existence of the asymptotic entropy and present three different, equivalent formulas, which are derived by three different techniques. In particular, we will show that the entropy is the rate of escape with respect to the Greenian metric. Moreover, we link asymptotic entropy with the rate of escape and volume growth resulting in two inequalities.

In this paper we prove a rate of escape theorem and a central limit theorem for isotropic random walks on Fuchsian buildings, giving formulae for the speed and asymptotic variance. In particular, these results apply to random walks induced by bi-invariant measures on Fuchsian Kac-Moody groups, however they also apply to the case where the building is not associated to any reasonable group structure. Our primary strategy is to construct a renewal structure of the random walk. For this purpose we define cones and cone types for buildings and prove that the corresponding automata in the building and the underlying Coxeter group are strongly connected. The limit theorems are then proven by adapting the techniques in [21]. The moments of the renewal times are controlled via the retraction of the walks onto an apartment of the building.

Abstract. We study certain phase transitions of branching random walks (BRW) on Cayley graphs of free products. The aim of this paper is to compare the size and structural properties of the trace, i.e., the subgraph that consists of all edges and vertices that were visited by some particle, with those of the original Cayley graph. We investigate the phase when the growth parameter λ is small enough such that the process survives but the trace is not the original graph. A first result is that the box-counting dimension of the boundary of the trace exists, is almost surely constant and equals the Hausdorff dimension which we denote by Φ(λ). The main result states that the function Φ(λ) has only one point of discontinuity which is at λc = R where R is the radius of convergence of the Green function of the underlying random walk. Furthermore, Φ(R) is bounded by one half the Hausdorff dimension of the boundary of the original Cayley graph and the behaviour of Φ(R) − Φ(λ) as λ ↑ R is classified.In the case of free products of infinite groups the end-boundary can be decomposed into words of finite and words of infinite length. We prove the existence of a phase transition such that if λ ≤λc the end boundary of the trace consists only of infinite words and if λ >λc it also contains finite words. In the last case, the Hausdorff dimension of the set of ends (of the trace and the original graph) induced by finite words is strictly smaller than the one of the ends induced by infinite words.