Philippe Biane

Some properties of crossings and partitions

We give a group theoretical interpretation of the lattice of non-crossing partitions of a cycle of length m. Using this interpretation, we prove that the group of skew-automorphisms of this lattice is isomorphic to the dihedral group D4m. We also give some generating functions, in terms of continued fraction expansions, for some statistics on the set of partitions involving the number of restricted crossings and the number of classes.

Littelmann paths and brownian paths

We study some path transformations related to Littelmann path model and their applications to representation theory and Brownian motion in a Weyl chamber.

Free diffusions, free entropy and free Fisher information

Abstract Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t =dS t − 1 2 f(X t ) dt where St is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as t→∞, towards the master field (i.e., the Gibbs state).

Quantum random walk on the dual of SU (n)

SummaryWe study a quantum random walk onA(SU(n)), the von Neumann algebra of SU(n), obtained by tensoring the basic representation of SU(n). Two classical Markov chains are derived from this quantum random walk, by restriction to commutative subalgebras ofA(SU(n)), and the main result of the paper states that these two Markov chains are related by means of Doobsh-processes.

Characters of symmetric groups and free cumulants

We investigate Kerov’s formula expressing the normalized irreducible characters of symmetric groups evaluated on a cycle, in terms of the free cumulants of the associated Young diagrams.

Poissonian Exponential Functionals, q-Series, q-Integrals, and the Moment Problem for log-Normal Distributions

Moments formulae for the exponential functionals associated with a Poisson process provide a simple probabilistic access to the so-called q-calculus, as well as to some recent works about the moment problem for the log-normal distributions.

Non-crossing cumulants of type B

We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu. Lattices of non-crossing partitions (of type A up to now) have played an important role in the combinatorics of free probability, primarily via the non-crossing cumulants of R. Speicher. Here we introduce the concept of non-crossing cumulant of type B; the inspiration for its definition is found by looking at an operation of restricted convolution of multiplicative functions, studied in parallel for functions on symmetric groups (in type A) and on hyperoctahedral groups (in type B). The non-crossing cumulants of type B live in an appropriate framework of non-commutative probability space of type B, and are closely related to a type B analogue for the R-transform of Voiculescu (which is the free probabilistic counterpart of the Fourier transform). By starting from a condition of vanishing of mixed cumulants of type B, we obtain an analogue of type B for the concept of free independence for random variables in a non-commutative probability space.

Continuous crystal and Duistermaat–Heckman measure for Coxeter groups

Abstract We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat–Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm–Liouville equations.

Minimal Factorizations of a Cycle and Central Multiplicative Functions on the Infinite Symmetric Group

We show that the number of factorizations?=?1??rof a cycle of lengthninto a product of cycles of lengthsa1, ?, ar, with ?rj=1(aj?1)=n?1, is equal tonr?1. This generalizes a well known result of J. Denes, concerning factorizations into a product of transpositions. We investigate some consequences of this result, for central multiplicative functions on the infinite symmetric group, and use them to give a new proof of a recent result of A. Nica and R. Speicher on non-crossing partitions.

Quelques proprietes du mouvement Brownien dans un cone

On montre plusieurs resultat generaux sur le comportement du mouvement Brownien dans un cone, conditionne a ne pas toucher le bord du cone. En particulier si le cone est un simplexe, on calcule la loi du minimum futur (pour lordre associe au cone) de ce processus. On applique ces resultats au cas dun cone douverture dans 2. Dans ce cas on a un analogue du theoreme de Pitman: si on considere le mouvement Brownien conditionne a ne pas toucher le bord, auquel on retranche trois fois son minimum futur, la composante de ce processus orthogonale a laxe du cone est un mouvement Brownien.