有限维几何上平均过程及其离散对偶的混合
Mixing of the Averaging process and its discrete dual on finite-dimensional geometries
摘要：
我们分析了由Aldous引入的平均过程的推广的$L^1$ -混合。这个过程发生在一个不断增长的序列图，我们假设是有限维的，在这个意义上，这些几何形状的随机游动满足一个家庭的纳什不等式。作为我们分析的副产品，我们提供了一个完整的图片的全变差混合的离散对偶的平均过程，我们称之为二项式分裂过程。这个过程的单个粒子本质上是底层图上的随机游动。当几个粒子一起进化时，当它们被放置在相邻的位置时，它们通过同步跳跃而相互作用。我们发现，给定$k$的粒子数和$n$的（增长）的大小的基础图形，系统表现出截断的全变差，如果$k\to\infty$和$k=O(n^2)$。最后，我们利用这两个过程之间的对偶性来证明二项式分裂满足Aldous的谱隙恒等式的一个版本，即该过程的弛豫时间与粒子的数量无关。
英文摘要：
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.
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