01.12.2023

Pierre Calka: Close-up on random convex interfaces

In this talk, we investigate random convex interfaces which are generated as convex hulls of random point sets. We are interested in their asymptotic behavior when the size of the input goes to infinity. In a first part, we mainly identify average and maximal fluctuations in the radial and longitudinal directions through precise convergence results. We observe that the model shares some common features with the famous KPZ universality class of certain growth processes, including the scaling of type 1:2:3 or the appearance of a limit distribution similar to the Tracy-Widom distribution. In a second part, we consider the peeling procedure which consists in iterating the construction of the convex hull of the point set. The so-called layers are asymptotically governed by a deterministic analytical model and we study the geometric characteristics of each of the first layers. The talk is based on several joint works with Joe Yukich and Gauthier Quilan.

Vitali Wachtel: Harmonic measure in a multidimensional gambler's problem

We consider a random walk in a truncated cone K_N , which is obtained by slicing cone K by a hyperplane at a growing level of order N. We study the behaviour of the Green function in this truncated cone as N increases. Using these results we also obtain the asymptotic behaviour of the harmonic measure. The obtained results are applied to a multidimensional gambler’s problem studied by Diaconis and Ethier (2022). In particular we confirm their conjecture that the probability of eliminating players in a particular order has the same exact asymptotic behaviour as for the Brownian motion approximation. We also provide a rate of convergence of this probability towards this approximation.