截断现象描述了一族遍历有限马氏链收敛到平衡点的急剧转变。许多自然家族的链被认为表现出切断，但建立这一事实往往是极具挑战性的。一类重要的链是$\G(n,d)$上的随机游动，$\G(n,d)$是$n$顶点上的随机$d$正则图。众所周知，几乎每一个这样的图$d\geq 3$是一个扩展，甚至本质上Ramanujan，这意味着混合时间的$O(\log n)$。根据Peres的猜想，对于这样的$d$，在$\G(n,d)$上的简单随机游走应该以高概率表现出截止。作为其特例，Durrett证明了随机3-正则图上的惰性随机游动的混合时间为w. h. p. $(6+o(1))\log_2 n$ .在这项工作中，我们证实了上述的结构，并建立截断的全变差，它的位置和它的最佳窗口，无论是简单的和非回溯随机游动$\G(n,d)$。也就是说，对于任何固定的$d\geq3$，$\G(n,d)$ w. h. p.上的简单随机游走在窗口顺序为$\sqrt{\log n}$的$\frac{d}{d-2}\log_{d-1} n$处截止。令人惊讶的是，$\G(n,d)$ w. h. p.上的非回溯随机游走已经在具有恒定窗口顺序的$\log_{d-1} n$处截止。我们进一步将这些结果推广到$\G(n,d)$，对于任何随$n$增长的$d=n^{o(1)}$（超过该值，混合时间为O（1）），我们在两个连续整数之一上建立混合时间的浓度。
英文摘要：
The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on $\G(n,d)$, a random $d$-regular graph on $n$ vertices. It is well known that almost every such graph for $d\geq 3$ is an expander, and even essentially Ramanujan, implying a mixing-time of $O(\log n)$. According to a conjecture of Peres, the simple random walk on $\G(n,d)$ for such $d$ should then exhibit cutoff with high probability. As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is w.h.p. $(6+o(1))\log_2 n$.
In this work we confirm the above conjectures, and establish cutoff in total-variation, its location and its optimal window, both for simple and for non-backtracking random walks on $\G(n,d)$. Namely, for any fixed $d\geq3$, the simple random walk on $\G(n,d)$ w.h.p. has cutoff at $\frac{d}{d-2}\log_{d-1} n$ with window order $\sqrt{\log n}$. Surprisingly, the non-backtracking random walk on $\G(n,d)$ w.h.p. has cutoff already at $\log_{d-1} n$ with constant window order. We further extend these results to $\G(n,d)$ for any $d=n^{o(1)}$ that grows with $n$ (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers.
We examine the mixing time for random walks on graphs. In particular we are interested on investigating graphs with bottlenecks. Furthermore, the cutoff phenomenon is examined.
Cutoff Phenomenon for Random Walks on Kneser Graphs
遍历马尔可夫链的截断现象描述了在一个可忽略的时间段内（称为截断窗口）收敛到其平稳分布的急剧转变。研究了Kneser图上简单随机游动的截断现象，Kneser图是一类遍历马氏链。给定两个整数$n$和$k$，Kneser图$K(2n+k,n)$定义为顶点集为$n$大小的$\{1,\ldots,2n+k\}$的所有子集，且如果$A\cap B =\emptyset$，则两个顶点$A$和$B$由一条边连接的图.我们表明，对于任何$k=O(n)$，$K(2n+k,n)$上的随机游走表现出一个截止在$\frac{1}{2}\log_{1+k/n}{(2n+k)}$与大小为$O(\frac{n}{k})$的窗口。
We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time $cn/2$, the curvature is asymptotically zero for $c\le 1$ an…
We consider the random walk on the hypercube which moves by picking an ordered pair $(i,j)$ of distinct coordinates uniformly at random and adding the bit at location $i$ to the bit at location $j$, modulo $2$. We show that this Markov chain has cutoff at time $\frac{3}{2}n\log n$ with window of size $n$, solving a question posed by Chung and Graham (1997).
We present an overview of the representation theoretic techniques used to study the mixing times of random walks on finite groups. We focus on the card shuffle studied by Diaconis and Shahshahani in the 1980s and a recent improvement on their result by Teyssier
A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees an…
We prove a cutoff for the random walk on random $n$-lifts of finite weighted graphs, even when the random walk on the base graph $\mathcal{G}$ of the lift is not reversible. The mixing time is w.h.p. $t_{mix}=h^{-1}\log n$, where $h$ is a constant associated to $\mathcal{G}$, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all $n$-lifts of $\mathcal{G}$. In the particular case where the base graph is a vertex with $d/2$ loops, $d$ even, we obtain a cutoff for a $d$-regular random graph (as did Lubetzky and Sly in \cite{cutoffregular} with a …
We consider dynamical percolation on the complete graph $K_n$, where each edge refreshes its state at rate $μ\ll 1/n$, and is then declared open with probability $p = λ/n$ where $λ> 1$. We study a random walk on this dynamical environment which jumps at rate $1/n$ along every open edge. We show that the mixing time of the full system exhibits cutoff at $\log n/μ$. We do this by showing that the random walk component mixes faster than the environment process; along the way, we control the time it takes for the walk to become isolated.
We consider Activated Random Walks on arbitrary finite networks, with particles being inserted at random and absorbed at the boundary. Despite the non-reversibility of the dynamics and the lack of knowledge on the stationary distribution, we explicitly determine the relaxation time of the process, and prove that separation cutoff is equivalent to the product condition. We also provide sharp estimates on the center and width of the cutoff window. Finally, we illustrate those results by establishing explicit separation cutoffs on various networks, including: (i) large finite subgraphs of any fix…
It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $d/ ((d-2)\log (d-1))\log n$. Such a bound is obtained by comparing the walk on $G_n$ to the walk on the infinite $d$-regular tree. If one can map another infinite transitive graph onto $G_n$, then we can improve the strategy by using a comparison with the random walk on this transitive graph (instead of that of the regular tree), and we obtain a lower bound of the form $1/h \log n$, where $h$ is the entropy rate associated with the …
The total-variation cutoff phenomenon has been conjectured to hold for simple random walk on all transitive expanders. However, very little is actually known regarding this conjecture, and cutoff on sparse graphs in general. In this paper we establish total-variation cutoff for simple random walk on Ramanujan complexes of type $\widetilde{A}_{d}$ $(d\geq1)$. As a result, we obtain explicit generators for the finite classical groups $\mathrm{PGL}_{n}(\mathbb{F}_{q})$ for which the associated Cayley graphs exhibit total-variation cutoff.
We show that for lazy simple random walks on finite spherically symmetric trees, the ratio of the mixing time and the relaxation time is bounded by a universal constant. Consequently, lazy simple random walks on any sequence of finite spherically symmetric trees do not exhibit pre-cutoff; this conclusion also holds for continuous-time simple random walks. This answers a question recently proposed by Gantert, Nestoridi, and Schmid. We also show that for lazy simple random walks on finite spherically symmetric trees, hitting times of vertices are (uniformly) non concentrated. Finally, we study t…
We investigate the asymptotic in $N$ of the mixing times of a Markov dynamics on $N-1$ ordered particles in an interval. This dynamics consists in resampling at independent Poisson times each particle according to a probability measure on the segment formed by its nearest neighbours. In the setting where the resampling probability measures are symmetric, the asymptotic of the mixing times were obtained and a cutoff phenomenon holds. In the present work, we focus on an asymmetric version of the model and we establish a cutoff phenomenon. An important part of our analysis consists in the derivat…
In this paper, we use the eigenvalues of the random to random card shuffle to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4} n \log n - \frac{1}{4}n\log\log{n}$ with window of order $n$, answering a conjecture of Diaconis.
For each $n,r \geq 0$, let $KG(n,r)$ denote the Kneser Graph; that whose vertices are labeled by $r$-element subsets of $n$, and whose edges indicate that the corresponding subsets are disjoint. Fixing $r$ and allowing $n$ to vary, one obtains a family of nested graphs, each equipped with a natural action by a symmetric group $\mathfrak{S}_n$, such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in [RW], while a systematic study of random walks on these families were considered in [RW2]. In this paper we illustrate that these ran…
In this thesis we introduce a new type of card shuffle called the one-sided transposition shuffle. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric group generated by a distribution which is non-constant on the conjugacy class of transpositions. Nevertheless, we provide an explicit formula for all eigenvalues of the shuffle by demonstrating a useful correspondence between eigenvalues and standard Young tableaux. This allows us to prove the existence of a total-variation cutoff…
It is recently proved by Lubetzky and Peres that the simple random walk on a Ramanujan graph exhibits a cutoff phenomenon, that is to say, the total variation distance of the random walk distribution from the uniform distribution drops abruptly from near $1$ to near $0$. There are already a few alternative proofs of this fact. In this note, we give yet another proof based on functional analysis and entropic consideration.
We establish universality of cutoff for simple random walk on a class of random graphs defined as follows. Given a finite graph $G=(V,E)$ with $|V|$ even we define a random graph $ G^*=(V,E \cup E')$ obtained by picking $E'$ to be the (unordered) pairs of a random perfect matching of $V$. We show that for a sequence of such graphs $G_n$ of diverging sizes and of uniformly bounded degree, if the minimal size of a connected component of $G_n$ is at least 3 for all $n$, then the random walk on $G_n^*$ exhibits cutoff w.h.p. This provides a simple generic operation of adding some randomness to a g…
The Diaconis--Gangolli random walk is an algorithm that generates an almost uniform random graph with prescribed degrees. In this paper, we study the mixing time of the Diaconis--Gangolli random walk restricted on $n\times n$ contingency tables over $\mathbb{Z}/q\mathbb{Z}$. We prove that the random walk exhibits cutoff at $\frac{n^2}{4(1- \cos{\frac{2 π}{q}})} \log n, $ when $\log q=o\left (\frac{\sqrt{\log n}}{\log \log n}\right )$.
Discovered in the context of card shuffling by Aldous, Diaconis and Shahshahani, the cutoff phenomenon has since then been established in a variety of Markov chains. However, proving cutoff remains a delicate affair, which requires a detailed knowledge of the chain. Identifying the general mechanisms underlying this phase transition -- without having to pinpoint its precise location -- remains one of the most fundamental open problems in the area of mixing times. In the present paper, we make a step in this direction by establishing cutoff for Markov chains with non-negative curvature, under a…
We show that the total-variation mixing time of the lamplighter random walk on fractal graphs exhibit sharp cutoff when the underlying graph is transient (namely of spectral dimension greater than two). In contrast, we show that such cutoff can not occur for strongly recurrent underlying graphs (i.e. of spectral dimension less than two).
In this paper, we are interested in the impact of communities on the mixing behavior of the non-backtracking random walk. We consider sequences of sparse random graphs of size $N$ generated according to a variant of the classical configuration model which incorporates a two-community structure. The strength of the bottleneck is measured by a parameter $α$ which roughly corresponds to the fraction of edges that go from one community to the other. We show that if $α\gg \frac{1}{\log N}$, then the non-backtracking random walk exhibits cutoff at the same time as in the one-community case, but with…
Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log |G|$ (ie $1 \ll k = |G|^{o(1)}$). A conjecture of Aldous and Diaconis (1985) asserts, for $k\gg\log|G|$, that the random walk on this graph exhibits cutoff. When $\log k \lesssim \log\log|G|$ (ie $k = (\log |G|)^{\mathcal O(1)}$), the only example of a non-Abelian group for which cutoff has been established is the dihedral group. We establish cutoff (as $p\to infty$) for the group of $d \times d$ unit upper triangular matrices with integer entries modulo…
We study the simple random walk on trees and give estimates on the mixing and relaxation time. Relying on a recent characterization by Basu, Hermon and Peres, we give geometric criteria, which are easy to verify and allow to determine whether the cutoff phenomenon occurs. We thoroughly discuss families of trees with cutoff, and show how our criteria can be used to prove the absence of cutoff for several classes of trees, including spherically symmetric trees, Galton-Watson trees of a fixed height, and sequences of random trees converging to the Brownian CRT.
We survey recent results concerning the total-variation mixing time of the simple exclusion process on the segment (symmetric and asymmetric) and a continuum analog, the simple random walk on the simplex with an emphasis on cutoff results. A Markov chain is said to exhibit cutoff if on a certain time scale, the distance to equilibrium drops abruptly from $1$ to $0$. We also review a couple of techniques used to obtain these results by exposing and commenting some elements of proof.
We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the forms $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n\log n$. We also show that this shuffle exhibits the cutoff phenomenon. In the appendix, we show that a similar random walk on the demihyperoctahedral group $D_n$ also has a cutoff at $\left(n-\frac{1}{2}\right)\log n$.
