NON-INTERSECTING LATTICE PATHS, CLASSICAL GROUP CHARACTERS, AND MULTIVARIATE HYPERGEOMETRIC SERIES

I shall review how non-intersecting lattice paths can be used as combinatorial models for classical group characters, and then show how these models provide "one-picture proofs" of identities for these characters, and for multivariate hypergeometric series. This is joint work with Richard Brent and Ole Warnaar.

A MULTIPLICATIVE ERGODIC THEOREM FOR P-ADIC LIE GROUPS

The celebrated Multiplicative Ergodic Theorem (MET) of Oseledets shows that under a finite first moment assumption the product of random real iid matrices behaves asymptotically like the sequence of powers of some fixed positive definite symmetric matrix. Kaimanovich observed that this property can be expressed in purely geometric terms using the symmetric space associated to GLn(R). This lead to the notion of a 'regular sequence' in a symmetric space, and by characterising these sequences in terms of spherical and horospherical coordinates Kaimanovich obtained a MET for noncompact semisimple real Lie groups with finite centre, generalising the original theorem of Oseledets.
In this talk we will discuss a p-adic analogue of this story. In this setting the symmetric space is replaced by the affine building of the p-adic group. We define regular sequences in affine buildings, and give a characterisation of these sequences in terms of analogues of the spherical and horospherical coordinates from the real theory. We then discuss applications to a MET for Lie groups defined over p-adic fields. This is joint with W. Woess.

WEIGHTED DEPENDENCY GRAPHS

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. I will explain how this works, and then present a more general notion of weighted dependency graphs. Applications to random graphs, subwords in texts generated by Markov sources, random Young diagrams and random walks in a cone will be given (the last two applications are still work in progress with J. Dousse and M. Thiel).

REDUCED EXPRESSIONS IN INFINITE COXETER GROUPS

Let (W,S) be an infinite Coxeter group with a finite set S of generators. The set Red(W) of reduced expressions is composed of all minimal-length words in S which represent the elements of W. Brink and Howlett showed in a celebrated paper that Red(W) is a "rational language" by constructing a finite state machine, or automaton, which accepts precisely the words of Red(W). This construction, which we will recall, relies on properties of the root system attached to W. In this talk we will introduce a new family of automata which all recognize Red(W), and whose definition involves only the weak order of W. We will also state two conjectures concerning the minimality of the various automata under consideration. This is joint work with Christophe Hohlweg and Nathan Williams.

RADIAL PARTS OF DIFFERENTIAL OPERATORS AND A ONE-PARAMETER FAMILY OF HYPERGEOMETRIC FUNCTIONS OF TYPE BC

We propose a generalization of the Matolci-Szucs uncertainty principle to a non-commutative locally compact group, where the measure of the support of the Fourier transform of a function is replaced by the integral of the rank of the corresponding operators. This is a joint work with Murad Ozaydin.

A NON-COMMUTATIVE, SHARP AND SYMMETRIC UNCERTAINTY INEQUALITY

We propose a generalization of the Matolci-Szucs uncertainty principle to a non-commutative locally compact group, where the measure of the support of the Fourier transform of a function is replaced by the integral of the rank of the corresponding operators. This is a joint work with Murad Ozaydin

LATTICE PATHS BELOW A LINE OF RATIONAL SLOPE

We analyze some enumerative and asymptotic properties of lattice paths below a line of rational slope. We illustrate our approach with Dyck paths under a line of slope 2/5. This answers Knuth's problem #4 from his Flajolet lecture during the conference Analysis of Algorithms (AofA'2014) in Paris in June 2014. Our approach extends the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths to the case of generating functions involving several dominant singularities. A key ingredient in the proof is the generalization of an old trick by Knuth himself (for enumerating permutations sortable by a stack), promoted by Flajolet and others as the kernel method. All the corresponding generating functions are algebraic, and they offer some new combinatorial identities, which can also be tackled in the A=B spirit of Wilf-Zeilberger-Petkovsek. We show how to obtain similar results for other slopes than 2/5. An interesting case being e.g. Dyck paths below the slope 2/3, which corresponds to the so called Duchon's club model. We also show how to use our techniques for lattice paths below an irrational slope and illustrate this with Dyck paths below y = x/sqrt(2).

ASYMPTOTIC GUESSES FOR 3D LATTICE WALKS IN THE POSITIVE OCTANT

The investigation of lattice walks restricted to a cone is a subject of ongoing interest. We provide some experimental results about asymptotic behavior for certain 3D lattice walks confined to the positive octant. In particular, we identify some models whose generating functions are most probably non-D-finite.

EXPONENTIAL FUNCTIONALS OF CONDITIONED LEVY PROCESSES AND LOCAL TIME OF A DIFFUSION IN A LEVY ENVIRONMENT

Exponential functionals of Levy processes have been widely studied over the past years and have multiple applications, among which the study of diffusions in random environment, the study of self-similar Markov processes or mathematical finance. We are interested in functionals of spectrally one-sided Levy processes conditioned to stay positive and establish some of their properties : finiteness, distribution tails, self-decomposability, smoothness of the density. We then apply these properties to the study of the asymptotic behavior of the local time of a diffusion in a spectrally negative Levy environment.

NEWTON TYPE POTENTIAL THEORY ASSOCIATED TO ROOT SYSTEMS

Let $R$ be a fixed root system on $R^d$ and $k$ be a nonnegative multiplicity function defined on $R$. Let $\Delta_k$ be the Dunkl-Laplace operator associated to $(R, k)$. We study the Dunkl-Newton kernel which is defined via the \Delta_k-heat kernel and the corresponding potential of a Radon measure on $R^d$. As application, we prove a Riesz type decomposition theorem for $\Delta_k$-subharmonic functions.

TAIL OF STATIONARY PROBABILITY OF STOCHASTIC DYNAMICAL SYSTEMS

A stochastic dynamical systems (SDS) is a random processes defined recursively by $X_n^x = \Psi_n(X_{n-1}^x)$, $X_0^x=x$, where $\Psi_n$ are i.i.d. random continuous transformations. We consider the class SDS on the real line, that are asymptotically linear in $+ \infty$ and $-\infty$, that includes interesting precess such as Affine recursion, Reflected random walk , Logistic recursion, AR(1)-model... . We investigate conditions for the existence of a stationary probability measure and describe the behaviour at infinity of such a measure.

KIRILLOV-FRENKEL CHARACTER FORMULA FOR LOOP GROUPS AND BROWNIAN SHEET

We will present in the framework of affine Lie algebra an analogue of the famous result which states that a real Brownian motion conditioned in Doob's sense to remain positive, is distributed as a Bessel 3 process. For this we will consider the coadjoint action of a Loop group of a compact group on the dual of the corresponding centrally extended Loop algebra and show that a Brownian motion in a Cartan subalgebra conditioned to remain in an affine Weyl chamber - which can be seen as a space time conditioned Brownian motion - is distributed as the radial part process of a Brownian sheet on the underlying Lie algebra. We will restrict ourself to the case of A_1^{(1)}.

ELEMENTARY APPROACH TO CONDITIONAL LOCAL LIMIT THEOREMS

I shall present a non-analytical method of proving local limit theorems for various conditional functionals of random walks. This method is based on a combination of classical (unconditioned) local limit theorems with conditional integral limit theorems, and allows to avoid the use of Fourier-transformations. I shall also give some combinatorial applications. (The talk is based on joint works with Denis Denisov and Martin Kolb.)

STOCHASTIC PROCESS ARISING FROM NONCOMMUTATIVE SYMETRIES

We will give some example of stochastic prosses arising from harmonic analysis and group representation theory; these include some random walks in cones, birth and death processes or some process related both to rank one symmetric spaces and to Heisenberg groups.

RANDOM WALKS ON NON-ARCHIMEDEAN GROUPS AND CHARACTERS

The Whittaker function was introduced by Jacquet and plays an important role in the study of Jacquet-Langlands L-functions. In particular, it appears in the Fourier expansion of (generalized) Eisenstein series. In this talk, I want to focus on a peculiar aspect of this function: in the case of GL_n(Q_p), it is exactly proportional to a Schur function (!).
I will present a probabilistic approach to this claim known as the Shintani-Casselman-Shalika formula. The general idea is as follows. After a few geometric considerations and some interesting potential theory, we reduce the problem of evaluating the probability that a lattice walk stays within a cone. This latter problem is a classical instance of Andre's Ballot theorem and the reflection principle

DUAL COMBINATORICS OF JACKS POLYNOMIALS

The characters of the symmetric groups could be alternatively defined completely without the representation theory as the coefficients in the expansion of Schur symmetric functions in the basis of power-sum symmetric functions. If we play the same game with Jack polynomials instead of Schur polynomials, we get some strange new quantities which are called Jack characters. Numerous computer experiments and some partial theoretical results suggest that Jack characters are related to some hypothetical one-parameter interpolation between the category of oriented maps (=graphs drawn on oriented surfaces) and the category of non-oriented maps.

Further reading:
Piotr Sniady. Top degree of Jack characters and enumeration of maps. http://arxiv.org/abs/1506.06361
Maciej Dolega, Valentin Féray, Piotr Sniady. Jack polynomials and orientability generating series of maps. Sem. Lothar. Combin., 70:Art. B70j, 50 pp., 2014

GIBBS MEASURES ON LITTELMANN PATHS MODEL

Littelmann paths are a central combinatorial object in the representation theory of semisimple Lie algebras: the concatenation of Littelmann paths gives the weight multiplicities and the decomposition into irreducible representations of arbitrary tensor products of representations. In this talk, I will describe the set of Gibbs measures for the Littelmann paths model coming from an irreducible representation of a simple Lie algebra: Gibbs measures are probability measures on infinite sequences of paths whose finite marginal laws are close to uniform distributions. I will consider the case of arbitrary Littelmann paths and the one of Littleman paths conditioned to stay in the Weyl chamber. The classification of the set of Gibbs measures has a probabilistic and an algebraic meaning which will be explained at the end of talk. Everything should be easily understandable for those who are not familiar with Littelmann paths models. This is a joint work with Cédric Lecouvey.