Errata corrige: orario di inizio ore 14:00 e non 14:15.
Data e ora: Martedì 8 Novembre, ore 14.00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Speaker: Matteo Quattropani, Sapienza Università di Roma
Title: Mixing of the Averaging process on graphs
Abstract: The Averaging process (a.k.a. repeated averages) is a mass redistribution model over the vertex set of a graph. Given a graph G, the process starts with a non-negative mass associated to each vertex. The edges of G are equipped with Poissonian clocks: when an edge rings, the masses at the two extremes of the edge are equally redistributed on these two vertices. Clearly, as time grows to infinity the state of the system will converge (in some sense) to a flat configuration in which all the vertices have the same mass. The process has been introduced to the probabilistic community by Aldous and Lanoue [1] in 2012, and recently received some attention thanks to the work of Chatterjee, Diaconis, Sly and Zhang [2], where the authors show an abrupt convergence to equilibrium (measured in L^1 distance) in the case in which the underlying graph is complete (and of diverging size). In this talk, I will present some recent results obtained in collaboration with F. Sau (IST Austria) [3,4] and P. Caputo (Roma Tre) [4]. In [3] we show that if the underlying graph is “finite dimensional” (e.g., a finite box of Z^d), then the convergence to equilibrium is smooth (i.e., without cutoff) when measured in L^p with p \in [1,2]. On the other hand, in [4] we show that a cutoff phenomenon (for the L^1 and L^2 distance to equilibrium) takes place when the underlying graph is the hypercube or the complete bipartite graph.
[1] David Aldous, and Daniel Lanoue. A lecture on the averaging process. Probab. Surv., 9:90–102, 2012. [2] Sourav Chatterjee, Persi Diaconis, Allan Sly, and Lingfu Zhang. A phase transition for repeated averages. Ann. Probab. 50(1):1–17, 2022. [3] Matteo Quattropani and Federico Sau. Mixing of the Averaging process and its discrete dual on finite-dimensional geometries. Ann. Appl. Probab. (to appear). [4] Pietro Caputo, Matteo Quattropani and Federico Sau. Cutoff for the Averaging process on the hypercube and complete bipartite graphs. (to appear).