我们研究随机环境中有限马尔可夫链的混合性质，该环境定义为确定性链与状态空间被均匀随机置换的链的混合。本文是配套论文的对应部分，在配套论文中我们专注于可逆模型，这使得证明中可以进行一些简化。我们在此考虑一般情形。在基础马尔可夫链的温和假设下，我们证明当链从典型状态出发时，以高概率，所得链在熵时间 $\log n / h$ 处表现出截止现象，其中 $h$ 是与链的熵相关的常数。然而，与可逆情形不同，熵时间的均匀截止并不成立，因为我们提供了一个例子，其中最坏情况混合时间至少具有多对数阶。我们还提供了最坏情况混合时间的多对数上界，这实际上在推导典型状态的主要结果中起着关键作用。顺便提及，我们的证明清晰地描绘了何时期望在熵时间出现均匀截止：它表现为证明中使用的覆盖马尔可夫链的均匀瞬态性质的结果，该覆盖链位于无限状态空间上，并投影回初始链。
英文摘要：
We investigate the mixing properties of a finite Markov chain in random environment defined as a mixture of a deterministic chain and a chain whose state space has been permuted uniformly at random. This work is the counterpart of a companion paper where we focused on a reversible model, which allowed for a few simplifications in the proof. We consider here the general case. Under mild assumptions on the base Markov chains, we prove that with high probability the resulting chain exhibits the cutoff phenomenon at entropic time $\log n / h$, $h$ being some constant related to the entropy of the chain, when the chain is started from a typical state. However contrary to the reversible case uniform cutoff at entropic time does not hold, as we provide an example where the worst-case mixing time has at least polylogarithmic order. We also provide a polylogarithmic upper bound on the worst-case mixing time, which in fact plays a crucial role in deriving the main result for typical states. Incidentally, our proof gives a clear picture of when to expect uniform cutoff at entropic time: it appears as the consequence of a uniform transience property for the covering Markov chain used throughout the proof, which lies on an infinite state space and projects back onto the initial chain.
Let $P$ be a bistochastic matrix of size $n$, and let $Π$ be a permutation matrix of size $n$. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by $Q=PΠ$. In other words, the chain alternates between random steps governed by $P$ and deterministic steps governed by $Π$. We show that if the permutation $Π$ is chosen uniformly at random, then under mild assumptions on $P$, with high probability, the chain $Q$ exhibits cutoff at time $\frac{\log n}{\mathbf{h}}$, where $\mathbf{h}$ is the entropic rate of $P$. Moreover, for deterministic permu…
Cutoff for mixtures of permuted Markov chains: reversible case
Département d’Informatique de l’ENS, École Normale Supérieure（巴黎高等师范学院计算机科学系）；CNRS（法国国家科学研究中心）；PSL Research University（巴黎文理研究大学）；INRIA Paris（法国国家信息与自动化研究所巴黎分部）；CNRS, I2M, Aix-Marseille Université（法国国家科学研究中心、马赛数学研究所、艾克斯-马赛大学）
Let $σ$ be a permutation of $\{0,\ldots,n\}$. We consider the Markov chain $X$ which jumps from $k\neq 0,n$ to $σ(k+1)$ or $σ(k-1)$, equally likely. When $X$ is at 0 it jumps to either $σ(0)$ or $σ(1)$ equally likely, and when $X$ is at $n$ it jumps to either $σ(n)$ or $σ(n-1)$, equally likely. We show that the identity permutation maximizes the expected hitting time of n, when the walk starts at 0. More generally, we prove that the hitting time of a random walk on a strongly connected $d$-directed graph is maximized when the graph is the line $[0,n]\cap\Z$ with $d-2$ self-loops at every verte…
We make a connection between the continuous time and lazy discrete time Markov chains through the comparison of cutoffs and mixing time in total variation distance. For illustration, we consider finite birth and death chains and provide a criterion on cutoffs using eigenvalues of the transition matrix.
We consider an $n$-tuple of independent ergodic Markov processes, each of which converges (in the sense of separation distance) at an exponential rate, and obtain a necessary and sufficient condition for the $n$-tuple to exhibit a separation cutoff. We also provide general bounds on the (asymmetric) window size of the cutoff, and indicate links to classical extreme value theory.
Given a permutation sigma of the integers {-n,-n+1,...,n} we consider the Markov chain X_{sigma}, which jumps from k to sigma (k\pm 1) equally likely if k\neq -n,n. We prove that the expected hitting time of {-n,n} starting from any point is Theta(n) with high probability when sigma is a uniformly chosen permutation. We prove this by showing that with high probability, the digraph of allowed transitions is an Eulerian expander; we then utilize general estimates of hitting times in directed Eulerian expanders.
In this article, we considers reversible Markov chains of which $L^2$-distances can be expressed in terms of Laplace transforms. The cutoff of Laplace transforms was first discussed by Chen and Saloff-Coste in [8], while we provide here a completely different pathway to analyze the $L^2$-distance. Consequently, we obtain several considerably simplified criteria and this allows us to proceed advanced theoretical studies, including the comparison of cutoffs between discrete time lazy chains and continuous time chains. For an illustration, we consider product chains, a rather complicated model wh…
A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of "worst" (in some sense) sets of stationary measure at least $α$, for some $α\in (0,1)$. We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some "worst" set of stationary measure at least $α$ was not hit by ti…
In this article, we consider products of ergodic Markov chains and discuss their cutoffs in the total variation. Through a new inequality relating the total variation and the Hellinger distance, we may identify the total variation cutoffs with cutoffs in the Hellinger distance. This provides a new scheme to study the total variation mixing of Markov chains, in particular, product chains. In the theoretical framework, a series of criteria are introduced to examine cutoffs and a comparison of mixing between the product chain and its coordinate chains is made in detail. For illustration, we consi…
We study convergence to equilibrium for a large class of Markov chains in random environment. The chains are sparse in the sense that in every row of the transition matrix $P$ the mass is essentially concentrated on few entries. Moreover, the random environment is such that rows of $P$ are independent and such that the entries are exchangeable within each row. This includes various models of random walks on sparse random directed graphs. The models are generally non reversible and the equilibrium distribution is itself unknown. In this general setting we establish the cutoff phenomenon for the…
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…
This paper studies Markov chains on the symmetric group $S_n$ where the transition probabilities are given by the Ewens distribution with parameter $θ>1$. The eigenvalues are identified to be proportional to the content polynomials of partitions. We show that the mixing time is bounded above by a constant depending only on the parameter if $θ$ is fixed. However, if it agrees with the number of permuted elements ($θ=n$), the sequence of chains has a total variation cutoff at $\frac{\log n}{\log 2}.$
Given a reversible Markov chain $P_n$ on $n$ states, and another chain $\tilde{P}_n$ obtained by perturbing each row of $P_n$ by at most $α_n$ in total variation, we study the total variation distance between the two stationary distributions, $\| π_n - \tildeπ_n \|$. We show that for chains with cutoff, $\| π_n - \tildeπ_n \|$ converges to $0$, $e^{-c}$, and $1$, respectively, if the product of $α_n$ and the mixing time of $P_n$ converges to $0$, $c$, and $\infty$, respectively. This echoes recent results for specific random walks that exhibit cutoff, suggesting that cutoff is the key property…
We study mixing of the Metropolis algorithm for a distribution on the hypercube that corresponds to the Erdős-Rényi random graph with edge probability p. This Markov chain has cutoff at max{p,1-p} n log n with window size n, a result proved by Diaconis and Ram (2000) using Fourier analysis. Here we give an alternative proof that relies on coupling and a projection to a two-dimensional Markov chain. This is done in the hope that probabilistic techniques will be easier to generalize to less symmetric distributions. We also describe a close relationship between the Metropolis and Gibbs samplers f…
We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times: our result show that these two continuous-time Markov chains display cutoff. Our analysis combines several ingredients including: a study of the hydrodynamic profile for ASEP, the use of monotonic eigenfunctions, stochastic comparisons and concentration inequalities.
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.
Consider a sequence of continuous-time irreducible reversible Markov chains and a sequence of initial distributions, $μ_n$. The sequence is said to exhibit $μ_n$-cutoff if the convergence to stationarity in total variation distance is abrupt, w.r.t. this sequence of initial distributions. In this work we give a characterization of $μ_n$-cutoff for an arbitrary sequence of initial distributions $μ_n$ (in the above setup). Our characterization is expressed in terms of hitting times of sets which are "worst" w.r.t. $μ_n$. Consider a Markov chain on $Ω$ whose stationary distribution in $π$. Let $t…
In the Gilbert-Shannon-Reeds shuffle, a deck of $N$ cards is cut into two approximately equal parts which are then riffled uniformly at random. Bayer and Diaconis famously showed that this Markov chain undergoes cutoff in total variation after $\frac{3\log(N)}{2 \log(2)}$ shuffles. We establish cutoff for the more general asymmetric riffle shuffles in which one cuts the deck into differently sized parts before riffling. The value of the cutoff point confirms a conjecture of Lalley from 2000. Some appealing consequences are that asymmetry always slows mixing and that total variation mixing is s…
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 analyze the convergence rates for a family of auto-regressive Markov chains $(X^{(n)}_k)_{k\geq 0}$ on $\mathbb R^d$, where at each step a randomly chosen coordinate is replaced by a noisy damped weighted average of the others. The interest in the model comes from the connection with a certain Bayesian scheme introduced by de Finetti in the analysis of partially exchangeable data. Our main result shows that, when $n$ gets large (corresponding to a vanishing noise), a cutoff phenomenon occurs.
The cutoff phenomenon describes a case where a Markov chain exhibits a sharp transition in its convergence to stationarity. In 1996, Diaconis surveyed this phenomenon, and asked how one could recognize its occurrence in families of finite ergodic Markov chains. In 2004, the third author noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing-time and spectral-gap tends to infinity, and conjectured that in many settings, this condition should also be sufficient. Diaconis and Saloff-Coste (2006) verified this conjecture for continuous-time b…
In this paper, we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as the one of birth-death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth-death one. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunfor…
We analyze a Markov chain, known as the product replacement chain, on the set of generating $n$-tuples of a fixed finite group $G$. We show that as $n \rightarrow \infty$, the total-variation mixing time of the chain has a cutoff at time $\frac{3}{2} n \log n$ with window of order $n$. This generalizes a result of Ben-Hamou and Peres (who established the result for $G = \mathbb{Z}/2$) and confirms a conjecture of Diaconis and Saloff-Coste that for an arbitrary but fixed finite group, the mixing time of the product replacement chain is $O(n \log n)$.
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…
Let $(X_t)_{t = 0 }^{\infty}$ be an irreducible reversible discrete time Markov chain on a finite state space $Ω$. Denote its transition matrix by $P$. To avoid periodicity issues (and thus ensuring convergence to equilibrium) one often considers the continuous-time version of the chain $(X_t^{\mathrm{c}})_{t \ge 0} $ whose kernel is given by $H_t:=e^{-t}\sum_k (tP)^k/k! $. Another possibility is to consider the associated averaged chain $(X_t^{\mathrm{ave}})_{t = 0}^{\infty}$, whose distribution at time $t$ is obtained by replacing $P$ by $A_t:=(P^t+P^{t+1})/2$. A sequence of Markov chains is…
The cutoff phenomenon describes the case when an abrupt transition occurs in the convergence of a Markov chain to its equilibrium measure. There are various metrics which can be used to measure the distance to equilibrium, each of which corresponding to a different notion of cutoff. The most commonly used are the total-variation and the separation distances. In this note we prove that the cutoff for these two distances are not equivalent by constructing several counterexamples which display cutoff in total-variation but not in separation and with the opposite behavior, including lazy simple ra…
A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity conditio…
Given a continuous time Markov Chain $\{q(x,y)\}$ on a finite set $S$, the associated noisy voter model is the continuous time Markov chain on $\{0,1\}^S$, which evolves in the following way: (1) for each two sites $x$ and $y$ in $S$, the state at site $x$ changes to the value of the state at site $y$ at rate $q(x,y)$; (2) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates $\{q(x,y)\}$ and the corresponding stationary distributions are almost uniform, then the mixing time has a sharp cutoff at time $\log|S|/2$ with a window of order 1. Lubetzky an…
We study the Swendsen-Wang dynamics for the $q$-state Potts model on the lattice. Introduced as an alternative algorithm of the classical single-site Glauber dynamics, the Swendsen-Wang dynamics is a non-local Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work we derive strong enough bounds on the mixing time, proving that the Swendsen-Wang dynamics on the lattice at sufficiently high temperatures exhibits a sharp transition from "unmixed" to "well-mixed," which is called the cutoff phenomenon. In particular, we establis…
We study the convergence rate to stationarity for a class of exchangeable partition-valued Markov chains called cut-and-paste chains. The law governing the transitions of a cut-and-paste chain are determined by products of i.i.d. stochastic matrices, which describe the chain induced on the simplex by taking asymptotic frequencies. Using this representation, we establish upper bounds for the mixing times of ergodic cut-and-paste chains, and under certain conditions on the distribution of the governing random matrices we show that the "cutoff phenomenon" holds.
The cutoff phenomenon for an ergodic Markov chain describes a sharp transition in the convergence to its stationary distribution, over a negligible period of time, known as cutoff window. We study the cutoff phenomenon for simple random walks on Kneser graphs, which is a family of ergodic Markov chains. Given two integers $n$ and $k$, the Kneser graph $K(2n+k,n)$ is defined as the graph with vertex set being all subsets of $\{1,\ldots,2n+k\}$ of size $n$ and two vertices $A$ and $B$ being connected by an edge if $A\cap B =\emptyset$. We show that for any $k=O(n)$, the random walk on $K(2n+k,n)…
The impulses, cutting entropy functional (EF) measure on trajectories Markov diffusion process, integrate information path functional (IPF) composing discrete information Bits extracted from observing random process. Each cut brings memory of the cutting entropy, which provides both reduction of the process entropy and discrete unit of the cutting entropy a Bit. Consequently, information is memorized entropy cutting in random observations which process interactions. The origin of information associates with anatomy creation of impulse enables both cut entropy and stipulate random process gener…
We establish three remarkable consequences of non-negative curvature for sparse Markov chains. First, their conductance decreases logarithmically with the number of states. Second, their displacement is at least diffusive until the mixing time. Third, they never exhibit the cutoff phenomenon. The first result provides a nearly sharp quantitative answer to a classical question of Ollivier, Milman and Naor. The second settles a conjecture of Lee and Peres for graphs with non-negative curvature. The third offers a striking counterpoint to the recently established cutoff for non-negatively curved …
An aperiodic and irreducible Markov chain on a finite state space converges to its stationary distribution. When convergence to equilibrium is measured by total variation distance, there exists an optimal coupling and a maximal coupling time. In this article, the maximal coupling time is compared to the hitting time of a specific state or set. Such sets, named halting sets, are studied in the case of symmetric birth-and-death chains and in some other examples. Some applications to the cutoff phenomenon are given. These results yield new methods to calculate cutoff times for some monotone birth…
We develop the theory of strong stationary duality for diffusion processes on compact intervals. We analytically derive the generator and boundary behavior of the dual process and recover a central tenet of the classical Markov chain theory in the diffusion setting by linking the separation distance in the primal diffusion to the absorption time in the dual diffusion. We also exhibit our strong stationary dual as the natural limiting process of the strong stationary dual sequence of a well chosen sequence of approximating birth-and-death Markov chains, allowing for simultaneous numerical simul…
We study the mixing time of the $(n,k)$ Bernoulli--Laplace urn model, where $k\in\{0,1,\ldots,n\}$. Consider two urns, each containing $n$ balls, so that when combined they have precisely $n$ red balls and $n$ white balls. At each step of the process choose uniformly at random $k$ balls from the left urn and $k$ balls from the right urn and switch them simultaneously. We show that if $k=o(n)$, this Markov chain exhibits mixing time cutoff at $\frac{n}{4k}\log n$ and window of the order $\frac{n}{k}\log\log n$. This is an extension of a classical theorem of Diaconis and Shahshahani who treated …
The aim of this paper is to develop a general theory for the class of skip-free Markov chains on denumerable state space. This encompasses their potential theory via an explicit characterization of their potential kernel expressed in terms of family of fundamental excessive functions, which are defined by means of the theory of Martin boundary. We also describe their fluctuation theory generalizing the celebrated fluctuations identities that were obtained by using the Wiener-Hopf factorization for the specific skip-free random walks. We proceed by resorting to the concept of similarity to iden…
For a given absorbing Markov chain $X^*$ on a finite state space, a chain $X$ is a sharp antidual of $X^*$ if the fastest strong stationary time of $X$ is equal, in distribution, to the absorption time of $X^*$. In this paper we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for Möbius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence - utilizing known results on a limiting di…
Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in $L^1$ on a system of size $n$ is $O(\log n)$. Whether in this regime there is cutoff, i.e. a sharp transition in the $L^1$-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at $(c+o(1))\log n$ for some fixed $c>0$, thus providing a rigorous stopping rule for this MCMC sampler. However, obtain…
