Cari colleghi,
riporto l'annuncio del seminario di Matteo Quattropani presso il
dipartimento di Matematica di Sapienza Università di Roma.
Cordiali saluti,
Lorenzo Taggi
*Data e ora: *Martedì 8 Novembre, ore 14.15, Sala di Consiglio,
Dipartimento di Matematica di Sapienza Università di Roma
*Title: *Mixing of the Averaging process on graphs
*Speaker:* Matteo Quattropani, Sapienza Università di Roma
*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).