Mixing of the Averaging process on finite dimensional geometries
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  in 2012, and recently received some attention thanks to the work of Chatterjee, Diaconis, Sly and Zhang , where the authors show an abrupt convergence to equilibrium (measured in L^1 norm) 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) . 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]. Finally, in the last part of the talk, I will show how such results on the mixing of the Averaging process imply sharp results on the mixing of a particle-analogue model.
- David Aldous, and Daniel Lanoue. A lecture on the Averaging process. Probab. Surv., 9:90-102, 2012.
- Sourav Chatterjee, Persi Diaconis, Allan Sly, and Lingfu Zhang. A phase transition for repeated averages. Ann. Probab. 50(1):1-17, 2022.
- Matteo Quattropani and Federico Sau. Mixing of the Averaging process and its discrete dual on finite-dimensional geometries. Ann. Appl. Probab. (to appear).