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

We analyze the $L^1$-mixing of a generalization of the Averaging process introduced by Aldous. The process takes place on a growing sequence of graphs which we assume to be finite-dimensional, 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 $k\to\infty$ 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.