arXiv subjects

Marc Peigné

Publications and source records attributed to Marc Peigné.

26 records · Page 2Linked to original sources

arXiv2012

Conditioned one-way simple random walk and representation theory

We call one-way simple random walk a random walk in the quadrant Z_+^n whose increments belong to the canonical base. In relation with representation theory of Lie algebras and superalgebras, we describe the law of such a random walk conditioned to stay in a closed octant, a semi-open octant or other types of semi-groups. The combinatorial representation theory of these algebras allows us to describe a generalized Pitman transformation which realizes the conditioning on the set of paths of the walk. We pursue here in a direction initiated by O'Connell and his coauthors [13,14,2], and also developed in [12]. Our work relies on crystal bases theory and insertion schemes on tableaux described by Kashiwara and his coauthors in [1] and, very recently, in [5].

math.RT↗

arXiv2010

Random walks in Weyl chambers and crystals

We use Kashiwara crystal basis theory to associate a random walk W to each irreducible representation V of a simple Lie algebra. This is achieved by endowing the crystal attached to V with a (possibly non uniform) probability distribution compatible with its weight graduation. We then prove that the generalized Pitmann transform defined by Biane, Bougerol and O'Connell for similar random walks with uniform distributions yields yet a Markov chain. When the representation is minuscule, and the associated random walk has a drift in the Weyl chamber, we establish that this Markov chain has the same law as W conditionned to never exit the cone of dominant weights. At the heart of our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice.

math.RT↗

arXiv2010

On some exotic Schottky groups

We construct a Cartan-Hadamard manifold with pinched negative curvature whose group of isometries possesses divergent discrete free subgroups with parabolic elements who do not satisfy the so-called "parabolic gap condition" . This construction relies on the comparaison between the Poincaré series of these free groups and the potential of some transfer operator which appears naturally in this context.

math.DS↗

arXiv2010

Autour de l'exposant critique d'un groupe kleinien

We present here some basic properties around the Poincaré exponent of a discrete group of isometries in pinched negatived curvature. We state some important results and present the main tools which are used in this domain.

math.GR↗

arXiv2010

On the horoboundary and the geometry of rays of negatively curved manifolds

In this paper we try to compare the "horoboundary" of a (not necessarily simply connected) negatively curved complete Riemannian manifold X with the visual one and describe with explicit examples some phenomenoms wich may appear when X is not simply connected.

math.DS↗

arXiv2010

Stochastic dynamical systems with weak contractivity properties (with a chapter featuring results of Martin Benda)

Consider a proper metric space X and a sequence of i.i.d. random continuous mappings F_n from X to X. It induces the stochastic dynamical system (SDS) X_n^x = F_n(X_{n-1}^x) starting at x in X. In this paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic iterations. We consider the case when the F_n are contractions and, in particular, discuss recurrence criteria and their sharpness for reflected random walk. In the second part, we consider the case where the F_n are Lipschitz mappings. The main results concern the case when the associated Lipschitz constants are log-centered. Prinicpal tools are the Chacon-Ornstein theorem and a hyperbolic extension of the space X as well as the process (X_n^x). The results are applied to the reflected affine stochastic recursion on the non-negative half-line.

math.PR↗

arXiv2009

On the growth of nonuniform lattices in pinched negatively curved manifolds

We study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices. These objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold .

math.DS↗

arXiv2006

On recurrence of reflected random walk on the half-line. With an appendix on results of Martin Benda

Let $(Y_n)$ be a sequence of i.i.d. real valued random variables. Reflected random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1} = |X_n - Y_{n+1}|$. In this note, we study recurrence of this process, extending a previous criterion. This is obtained by determining an invariant measure of the embedded process of reflections.

math.PR↗