Related papers: A technical report on hitting times, mixing and cu…
Consider a Markov chain with finite state space and suppose you wish to change time replacing the integer step index $n$ with a random counting process $N(t)$. What happens to the mixing time of the Markov chain? We present a partial reply…
We introduce the concept of a Markov influence system (MIS) and analyze its dynamics. An MIS models a random walk in a graph whose edges and transition probabilities change endogenously as a function of the current distribution. This…
In this study, we consider sequences drawn from time-homogeneous Markov chains and introduce a novel approach for estimating first hitting-time distributions to specified terminal states. Our method- ology is based on the…
We study random walks on the giant component of the Erd\H{o}s-R\'enyi random graph ${\cal G}(n,p)$ where $p=\lambda/n$ for $\lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently…
We define the hitting time for a Markov decision process (MDP). We do not use the hitting time of the Markov process induced by the MDP because the induced chain may not have a stationary distribution. Even it has a stationary distribution,…
Let $(X_t)$ be a discrete time Markov chain on a general state space. It is well-known that if $(X_t)$ is aperiodic and satisfies a drift and minorization condition, then it converges to its stationary distribution $\pi$ at an exponential…
Given a labelled tournament on $[n]$, \emph{inverting} a vertex subset $X$ means reversing every edge with both endpoints in $X$. Alon, Powierski, Savery, Scott, and Wilmer~\cite{AlonPowierskiSaveryScottWilmer2024} asked for the mixing time…
In classical probability theory, the term "cutoff" describes the property of some Markov chains to jump from (close to) their initial configuration to (close to) completely mixed in a very narrow window of time. We investigate how coherent…
The basic question in perturbation analysis of Markov chains is: how do small changes in the transition kernels of Markov chains translate to chains in their stationary distributions? Many papers on the subject have shown, roughly, that the…
This article describes an accurate procedure for computing the mean first passage times of a finite irreducible Markov chain and a Markov renewal process. The method is a refinement to the Kohlas, Zeit fur Oper Res, 30,197-207, (1986)…
The mixing time of a Markov chain determines how fast the iterates of the Markov chain converge to the stationary distribution; however, it does not control the dependencies between samples along the Markov chain. In this paper, we study…
Slow mixing is the central hurdle when working with Markov chains, especially those used for Monte Carlo approximations (MCMC). In many applications, it is only of interest to estimate the stationary expectations of a small set of…
We consider Markov chains with random transition probabilities which, moreover, fluctuate randomly with time. We describe such a system by a product of stochastic matrices, $U(t)=M_t\cdots M_1$, with the factors $M_i$ drawn independently…
We find the total variation mixing time of the interchange process on the dumbbell graph (two complete graphs, $K_n$ and $K_m$, connected by a single edge), and show that this sequence of chains exhibits the cutoff phenomenon precisely when…
We consider the zero-range process with arbitrary bounded monotone rates on the complete graph, in the regime where the number of sites diverges while the density of particles per site converges. We determine the asymptotics of the mixing…
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…
We define the spectral gap of a Markov chain on a finite state space as the second-smallest singular value of the generator of the chain, generalizing the usual definition of spectral gap for reversible chains. We then define the relaxation…
In this article we study a small random perturbation of a linear recurrence equation. If all the roots of its corresponding characteristic equation have modulus strictly less than one, the random linear recurrence goes exponentially fast to…
For general spin systems, we prove that a contractive coupling for any local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dynamics,…
For any discrete target distribution, we exploit the connection between Markov chains and Stein's method via the generator approach and express the solution of Stein's equation in terms of expected hitting time. This yields new upper bounds…