Bruno Schapira

A Local Limit Theorem for Random Walks on the Chambers of ˜ A 2 Buildings

In this paper we outline an approach for analysing random walks on the chambers of buildings. The types of walks that we consider are those which are well adapted to the structure of the building: Namely walks with transition probabilities p(c, d) depending only on the Weyl distance d(c, d). We carry through the computations for thick locally finite affine buildings of type A2 to prove a local limit theorem for these buildings. The technique centres around the representation theory of the associated Hecke algebra. This representation theory is particularly well developed for affine Hecke algebras, with elegant harmonic analysis developed by Opdam ([28], [29]). We give an introductory account of this theory in the second half of this paper.

Random walk on a building of type Ãr and Brownian motion of the Weyl chamber

In this paper we study a random walk on an affine building of type

Moderate deviations for the range of a transient random walk: path concentration

\tilde{A}_r

Windings of planar random walks and averaged Dehn function

, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane \cite{Bia2}. This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one \cite{BJ}. The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in \cite{AST}.

Strongly Vertex-Reinforced-Random-Walk on a complete graph

We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the {\it Swiss cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.

Random walks with occasionally modified transition probabilities

We prove a sharp estimate on the expected value of the integral of the index of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.