Recently, random walks on Hecke algebras were recognized by A. Bufetov as a natural framework for the study of multi-species interacting particle systems. As a corollary, the Mallows measure can be viewed as the universal stationary blocking measure of interacting particle systems arising from random walks on Hecke algebras. Furthermore, the involution in Hecke algebras implies the color-position symmetry, which is a powerful tool for the asymptotic analysis of multi-species interacting particle systems. In this talk, we explore two facets of random walks on Hecke algebras. The first part focuses on the asymptotic behavior of the Mallows measure. In the second part, we consider applications of the color-position symmetry, particularly in the context of shock fluctuations in the half-line open Totally Asymmetric Simple Exclusion Process (TASEP) and Asymmetric Simple Exclusion Process (ASEP).

We investigate the topology and local geometry of various models of random cubical complexes. In the first part, we consider two types of random subcomplexes of the regular cubical grid: percolation clusters and the Eden cell growth model, analyzing their geometric and topological features. In the second part, we study the fundamental group of random 2-dimensional subcomplexes of an n-dimensional cube, identifying interesting threshold phenomena. This latter model serves as a cubical analogue of the Linial–Meshulam model for random simplicial complexes.

The gene regulatory network models all the biochemical reactions between the different species (proteins, mRNA, etc.) present in a cell. A stochastic approach to this network, using jump processes, has been studied during the '70s (Kurtz), and it was established that in large populations, the rescaled process (thought as a concentrations process) converges towards the solution of an EDO, which is entirely deterministic. In order to obtain a random limit, Crudu, Debussche and Radulescu proposed a multiscale model. In this talk, I will propose a relevant stochastic representation of the multiscale model, for which it is possible to obtain the uniform convergence towards a PDMP and a CLT for the fluctuations around this limit.

Stochastic differential equations (SDE) of McKean-Vlasov type or mean-field type are SDE where the solution's law appears inside the equation's coefficients making them more complicated to solve. On the other hand, such equations have been prominently applied in finance, agent dynamics and machine learning. In this mini course we offer an introduction to this framework covering wellposedness of McKean-Vlasov SDE and properties alongside the associated approximating interacting SDE particle systems and corresponding propagation of chaos.

Stochastic differential equations (SDE) of McKean-Vlasov type or mean-field type are SDE where the solution's law appears inside the equation's coefficients making them more complicated to solve. On the other hand, such equations have been prominently applied in finance, agent dynamics and machine learning. In this mini course we offer an introduction to this framework covering wellposedness of McKean-Vlasov SDE and properties alongside the associated approximating interacting SDE particle systems and corresponding propagation of chaos.