98 research outputs found
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 Poincar\'e series of these
free groups and the potential of some transfer operator which appears naturally
in this context
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].Comment: 32 page
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.Comment: The second version presents minor modifications to the previous on
Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths
We introduce and characterize central probability distributions on Littelmann
paths. Next we establish a law of large numbers and a central limit theorem for
the generalized Pitmann transform. We then study harmonic functions on
multiplicative graphs defined from the tensor powers of finite-dimensional Lie
algebras representations. Finally, we show there exists an inverse of the
generalized Pitman transform defined almost surely on the set of infinite paths
remaining in the Weyl chamber and explain how it can be computed.Comment: 27 pages, minor corrections and a simpler definition of the Pitman
invers
ON RECURRENCE OF REFLECTED RANDOM WALK ON THE HALF-LINE
Abstract. Let (Yn) be a sequence of i.i.d. real valued random variables. Reflected random walk (Xn) is defined recursively by X0 = x ≥ 0, Xn+1 = |Xn − Yn+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
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