Universality is one of the central questions in many probability models and its importance is well established from classical results like CLT and LLN to the so called KPZ equation. In this talk, we explore universality in first passage percolation in the plane. We show that this model is not universal when we consider measures that are ergodic, but not necessarily i.i.d.. To prove this, we construct a family of measures and explicitly compute the variant and wandering exponents. To the best of our knowledge, this is the first example where these two important quantities can be computed. Based on joint work with Christopher Hoffman.

Modelling relationships between individuals is a classical question in social sciences and clustering individuals according to the observed patterns of interactions allows to uncover a latent structure in the data. The stochastic block model is a popular approach for grouping individuals with respect to their social comportment. When several relationships of various types can occur jointly between individuals, the data are represented by multiplex networks where more than one edge can exist between the nodes. We extend stochastic block models to multiplex networks to obtain a clustering based on more than one kind of relation- ship. We propose to estimate the parameters—such as the marginal probabilities of assignment to groups (blocks) and the matrix of probabilities of connections between groups—through a variational expectation–maximization procedure. Consistency of the estimates is studied. The number of groups is chosen by using the integrated completed likelihood criterion, which is a penalized likelihood criterion. Multiplex stochastic block models arise in many situations but our applied example is motivated by a network of French cancer researchers. This is a joint work with Pierre Barbillon (AgroParisTech) and Sophe Donnet (Inra).

We consider a class of strictly positive martingales satisfying dS(t) = A(t)S(t)dW(t) where W is a Brownian motion and the process A is a stochastic process independent of W. We derive various results of the form E f(S(T)) = E F[S] , where E denotes expectation, F[S] is a functional of the path of S (e.g., possibly depending jointly on the running maximum, running minimum and quadratic variation of S) and the function f is determined from F. From a financial standpoint, these results allow us to determine the value of a path-dependent claim on a stock S relative to value of a path-independent European claim. This is joint work with Peter Carr and Roger Lee.

We study a process given by the first n coordinates of a Brownian motion on the unit sphere in the (n+l)-dimensional Euclidean space and obtain a stochastic differential equation (SDE) it satisfies. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. The uniqueness results and the skew-product decomposition are considered and proved for a more general SDE on the unit ball in the n-dimensional Euclidean space. The skew-product decomposition suggests a simulation algorithm for the increments of the spherical Brownian motion and a simplification reduces it to the simulation of Wright-Fisher diffusions. We use a recent algorithm for an exact simulation of Wright-Fisher diffusions given by Jenkins & Spanò (2017) to obtain an efficient algorithm for the increments of spherical Brownian motion allowing a wide range of time-steps.

We study random surfaces associated with random Young tableaux. When the tableau is rectangular, the existence of a limiting surface was proved by Pittel-Romik. We show that the fluctuations away from this limit are gausssian in the corners and Tracy-Widom on the edge. When the tableau is triangular, we study the fluctuations on the corners and show limit laws which are related to Mittag-Leffler distributions. Joint work with Cyril Banderier and Michael Wallner.

En esta plática voy a presentar un poco sobre los equilibrios de Nash para juegos diferenciales estocásticos. En particular, estudiaremos estos juegos asociando un problema de control óptimo de tal forma que una solución óptima de este problema será un equilibrio de Nash para el juego original. A esta técnica la llamamos el enfoque potencial. Gracias a este enfoque simplificamos el problema de existencia de equilibrios puros. Sin embargo, ¿será posible establecer alguna propiedad de estabilidad para esos equilibrios?

I will discuss the problem of computing the eigenvalues distribution of polynomials in random matrices, in the limit where the size of the matrices goes to infinity. In this context, Voiculescu's Free Probability Theory gives analytic tools to consider this question when the random matrices are in "generic position", in particular when they are invariant by conjugation by unitary matrices. Here we work under a much weaker assumption, assuming only that the random matrices are invariant in law by conjugation by permutation matrices. This requires a more general method, known as Traffic Probability Theory. Since recently, with this approach we were only able to give a combinatorial description for the moments of the limit eigenvalues distribution. More recently, we discovered that freeness in the sense of traffics implies Voiculescu's notion of freeness with amalgamation over the diagonal. In particular, this yields new numerical methods to compute limiting eigenvalues distributions.

Se revisita el teorema de M. Wschebor sobre los pequeños incrementos de procesos que verifican propiedades de invarianza por cambio de escala y de estacionariedad y se deducen Principios de Grandes Desviaciones. En colaboración con José León (Montevideo).

TBA