We establish exact formulae for the (positivity) persistence probabilities of an autoregressive sequence with symmetric uniform innovations in terms of certain families of polynomials, most notably a family introduced by Mallows and Riordan as enumerators of finite labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is also discussed. Two further results provide factorizations of general autoregressive models, one for negative drifts with continuous innovations, and one for positive drifts with continuous and symmetric innovations. The second factorization extends a classical universal formula of Sparre Andersen for symmetric random walks. Our results also lead to explicit asymptotic estimates for the persistence probabilities. This is a joint work with Gerold Alsmeyer, Alin Bostan and Thomas Simon (Adv. Appl. Math., 2023). 

Can we reconstruct a directed acyclic graph having only access to its v-structures, encoding conditional independence between the sites, but without knowing its edge directions? In this talk, we study the probability to have a unique way of such a reconstruction when the directed acyclic graph G is chosen uniformly at random on a fixed number of sites. More generally, we study the size of its Markov equivalence class, containing all graphs with the same edge set as G when forgetting the edge directions, and having the same v-structures. This talk is based on joint work with Allan Sly (Princeton University). 

We study a class of kinetic-type differential equations $u_t+u=Qu$, where $Q$ is an inhomogeneous smoothing transform. We show that under mild assumptions on $Q$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to\infty$. The talk is based on a joint work with Dariusz Buraczweski and Alexander Marynych

Let A and B be large unitarily invariant independent Hermitian random matrices whose 

empirical spectra approximate probability measures $\mu$ and $\nu$ on the real line. The central results of free probability says that the empirical spectrum of the random sum $A+B$ approximates a certain measure $\mu \boxplus \nu$ on the real line, known as the additive free convolution of $\mu$ and $\nu$. 

We apply a large deviation principle on the symmetric group to a recent formula of Marcus, Spielman and Srivastava to prove a new variational description of the additive free convolution.

We prove similar formulas for other operations in free probability, as well as new inequalities relating free and classical operations.

This is joint work with Octavio Arizmendi (CIMAT, Mexico).