Mixing of the Averaging process and its discrete dual on finite-dimensional geometries

Abstract

We analyze the L 1 -mixing of a generalization of the Averaging process introduced by Aldous [Ald11]. The process takes place on a growing sequence of graphs which we assume to be finitedimensional, in the sense that the random walk on those geometries satisfies a family of Nash inequalities. As a byproduct of our analysis, we provide a complete picture of the total variation mixing of a discrete dual of the Averaging process, which we call Binomial Splitting process. A single particle of this process is essentially the random walk on the underlying graph. When several particles evolve together, they interact by synchronizing their jumps when placed on neighboring sites. We show that, given k the number of particles and n the (growing) size of the underlying graph, the system exhibits cutoff in total variation if k → ∞ and k = O(n 2 ). Finally, we exploit the duality between the two processes to show that the Binomial Splitting satisfies a version of Aldous' spectral gap identity, namely, the relaxation time of the process is independent of the number of particles.

Repeated Averages on Graphs

“…All our existing results support Conjecture 1. Our result on cycles resolve a conjecture in Spiro [2021], independently from Quattropani and Sau [2021], and with different techniques.…”

“…In order to establish our results we make several observations about the process, such as the worst case initialization is always a standard basis vector. Our results add to the body of work of Aldous [1989], Aldous and Lanoue [2012], Quattropani and Sau [2021], Cao [2021], Olshevsky and Tsitsiklis [2009], and others. The renewed interest is due to an analogy to a question related to the Google's supremacy circuit.…”

“…In [Chatterjee et al, 2021], the authors prove a cut-off phenomenon for the complete graph. It is also proved in Quattropani and Sau [2021] that certain families of graphs do not exhibit cut-offs. Are there other graphs for which one can prove or disprove a cut-off phenomenon, for instance for star graphs?…”

“…They succeeded in showing a sharp cutoff phenomenon answering a conjecture of Jean Bourgain and a question of Ramis Movassagh. They proved that the L 1 convergence happens in time 1 2 log 2 n log n and the cutoff window is of magnitude n √ log n. Quattropani and Sau [2021] considers a dual problem to averaging and characterizes the L 1 mixing time. They analyze the cutoff phenomena with respect to the number of random walkers on the graph.…”

Entropy inequalities for random walks and permutations

“…The special case when particles are unlabeled and α A = 0 unless |A| = 2 is sometimes referred to as the binomial splitting model. The latter has been recently studied in [39], where the independence on the number of particles for the spectral gap was obtained by a different argument. As discussed in [39], by duality, controlling the convergence to equilibrium for this model allows one to control the approach to stationarity for the averaging processes introduced in [2].…”