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.
Il seminario si svolgerà in presenza presso il Dipartimento di Matematica e
Fisica - Lungotevere Dante 376 (Blocco Aule) - *Aula M2*