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Random Walks and Trees

Zhan Shi (2011)

ESAIM: Proceedings

These notes provide an elementary and self-contained introduction to branching random walks. Section 1 gives a brief overview of Galton–Watson trees, whereas Section 2 presents the classical law of large numbers for branching random walks. These two short sections are not exactly indispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-goût to the main part, Section 3, where branching random walks...

Random walks in ( ℤ + ) 2 with non-zero drift absorbed at the axes

Irina Kurkova, Kilian Raschel (2011)

Bulletin de la Société Mathématique de France

Spatially homogeneous random walks in ( ℤ + ) 2 with non-zero jump probabilities at distance at most 1 , with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes.

Random Walks in Attractive Potentials: The Case of Critical Drifts

Dmitry Ioffe, Yvan Velenik (2010)

Actes des rencontres du CIRM

We consider random walks in attractive potentials - sub-additive functions of their local times. An application of a drift to such random walks leads to a phase transition: If the drift is small than the walk is still sub-ballistic, whereas the walk is ballistic if the drift is strong enough. The set of sub-critical drifts is convex with non-empty interior and can be described in terms of Lyapunov exponents (Sznitman, Zerner ). Recently it was shown that super-critical drifts lead to a limiting...

Random walks on finite rank solvable groups

Ch. Pittet, Laurent Saloff-Coste (2003)

Journal of the European Mathematical Society

We establish the lower bound p 2 t ( e , e ) ≿ exp ( − t 1 / 3 ) , for the large times asymptotic behaviours of the probabilities p 2 t ( e , e ) of return to the origin at even times 2 t , 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 .)

Random walks on free products

M. Gabriella Kuhn (1991)

Annales de l'institut Fourier

Let G = * j = 1 q + 1 G n j + 1 be the product of q + 1 finite groups each having order n j + 1 and let μ be the probability measure which takes the value p j / n j on each element of G n j + 1 ∖ { e } . In this paper we shall describe the point spectrum of μ in C reg * ( G ) and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers n j . We also compute the continuous spectrum of μ in C reg * ( G ) in several cases. A family of irreducible representations of G , parametrized on the continuous spectrum of μ ,...

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