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Green functions for killed random walks in the Weyl chamber of Sp(4)

Kilian Raschel — 2011

Annales de l'I.H.P. Probabilités et statistiques

We consider a family of random walks killed at the boundary of the Weyl chamber of the dual of Sp(4), which in addition satisfies the following property: for any ≥ 3, there is in this family a walk associated with a reflection group of order 2. Moreover, the case = 4 corresponds to a process which appears naturally by studying quantum random walks on the dual of Sp(4). For all the processes belonging to this family, we find the exact asymptotic of the Green functions along all infinite paths of...

Counting walks in a quadrant: a unified approach via boundary value problems

Kilian Raschel — 2012

Journal of the European Mathematical Society

The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations...

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

Irina KurkovaKilian Raschel — 2011

Bulletin de la Société Mathématique de France

Spatially homogeneous random walks in ( + ) 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.

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