Martedì 14 Maggio 2024 alle ore 14:30, Federico Sau (Università degli Studi di Trieste) terrà il seminario di Probabilità dal titolo 

"Scaling limits of the averaging process".

Abstract:
The averaging process on a graph is a continuous-space Markov chain, which is commonly interpreted as an opinion dynamics, a distributed algorithm, or an interface moving through a randomized sequence of deterministic local updates. Its dynamics goes as follows. Attach i.i.d. Poisson clocks to edges, and assign real values to vertices; at the arrival times of these clocks, update the values with their average. As time runs, the averaging process converges to a flat configuration, and one major problem in the field is that of quantifying the speed of convergence to its degenerate equilibrium in terms of characteristic features of the underlying graph. In this talk, after reviewing some basic properties and recent results on mixing times for the averaging process on general graphs, we focus on the discrete $d$-dimensional torus, and on some finer properties of the process in this setting. We discuss some quantitative features (e.g., limit profile, early concentration and local smoothness), and look at nonequilibrium fluctuations, a particularly interesting problem in this degenerate context lacking a non-trivial notion of local equilibrium. If time permits, we will touch on the main ideas of the proof of such scaling limits, which combine tools from Malliavin calculus in Poisson space, their probabilistic dynamic interpretations, and some new discrete-gradient estimates. Talk based on the preprints arXiv.2311.14176, arXiv.2403.02032.