Cayley 图混合时间的敏感性

Sensitivity of mixing times in Eulerian digraphs

Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows. However, we establish that for Eulerian digraphs, the mixing time is $O(mn)$, where $m$ is the number of edges and $n$ is the number of vertices. In the reversible case, the mixing time is robust to the change of the laziness parameter. Surprisingly, in the directed setting the mixing time can be sensitive to such changes. We also study exploration and cove…

On sensitivity of mixing times and cutoff

A sequence of chains exhibits (total-variation) cutoff (resp., pre-cutoff) if for all $0<ε< 1/2$, the ratio $t_{\mathrm{mix}}^{(n)}(ε)/t_{\mathrm{mix}}^{(n)}(1-ε)$ tends to 1 as $n \to \infty $ (resp., the $\limsup$ of this ratio is bounded uniformly in $ε$), where $t_{\mathrm{mix}}^{(n)}(ε)$ is the $ε$-total-variation mixing-time of the $n$th chain in the sequence. We construct a sequence of bounded degree graphs $G_n$, such that the lazy simple random walks (LSRW) on $G_n$ satisfy the "product condition" $\mathrm{gap}(G_n) t_{\mathrm{mix}}^{(n)}(ε) \to \infty $ as $n \to \infty$, where $\mat…

Upper bounds on mixing time of finite Markov chains

We provide a general framework for computing upper bounds on mixing times of finite Markov chains when its minimal ideal is left zero. Our analysis is based on combining results by Brown and Diaconis with our previous work on stationary distributions of finite Markov chains. Stationary distributions can be computed from the Karnofsky--Rhodes and McCammond expansion of the right Cayley graph of the finite semigroup underlying the Markov chain. Using loop graphs, which are planar graphs consisting of a straight line with attached loops, there are rational expressions for the stationary distribut…

幂零群随机Cayley图的截止现象

Cutoff for random Cayley graphs of nilpotent groups

研究秩和幂零类有界的有限幂零群随机Cayley图上简单随机游走的截止现象，证明随机生成元情形下高概率出现截止，并给出一般生成元集合下谱隙和混合时间与阿贝尔化投影渐近等价的判据。

Dull cut off for circulants

Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phenomenon has been shown to occur in families such as random walks on a hypercube [1] in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion [4]. Our examples agree with …

Mallows permutation models with $L^1$ and $L^2$ distances I: hit and run algorithms and mixing times

Mallows permutation model, introduced by Mallows in statistical ranking theory, is a class of non-uniform probability measures on the symmetric group $S_n$. The model depends on a distance metric $d(σ,τ)$ on $S_n$, which can be chosen from a host of metrics on permutations. In this paper, we focus on Mallows permutation models with $L^1$ and $L^2$ distances, respectively known in the statistics literature as Spearman's footrule and Spearman's rank correlation. Unlike most of the random permutation models that have been analyzed in the literature, Mallows permutation models with $L^1$ and $L^2$…

Nilprogressions and groups with moderate growth

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a number of applications to the geometry and spectrum of finite Cayley graphs. For example, we show that a finite group has moderate growth in the sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a fixed power of the cardinality of the group. We call such groups almost flat and show that they have a subgroup of bounded index admitti…

圆柱上的扩散限制聚集

Diffusion Limited Aggregation on a Cylinder

本文考虑圆柱G × N上的DLA过程。结果表明，这个过程“增长武器”，提供的基础图G有足够小的混合时间。具体地，如果G的混合时间至多为（log |G|）^（2- \eps），则簇到达圆柱体的第m层所花费的时间至多为m阶|G| /loglog |G|。特别是，我们得到的例子无限凯莱图度5，这些图上的DLA集群具有任意小的密度。此外，我们提供了一个上界的“武器”增长的速度。这个界限是有效的一个大类的基图G，包括离散环面的尺寸至少为3。它还表明，对于任何基图G，密度的DLA过程中的G-柱有关的速率，在该集团的臂增长。这意味着，对于任何点传递的G，G-柱上的DLA的密度是有界的2/3。