This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary… Expand

We consider a class of pure jump Markov processes in Rd whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are… Expand

We consider symmetric Markov chains on the integer lattice in d dimensions, where a ∈ (0, 2) and the conductance between x and y is comparable to |x-y| -(d+α) . We establish upper and lower bounds… Expand

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie–Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity… Expand

Suppose that a coin with bias θ is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let μ θ be the distribution of the observed sequence of coin tosses, and… Expand

This article provides the first procedure for computing a fully data-dependent interval that traps the mixing time $t_{\text{mix}}$ of a finite reversible ergodic Markov chain at a prescribed confidence level.Expand

The existence of partial Hadamard matrices can be proved by showing that there is positive probability of a random walk returning to the origin after a specified number of steps.Expand

We can determine the biases {θ(z)}z∈Z, using only the outcomes of these coin tosses and no information about the path of the random walker, up to a shift and reflection of Z.Expand

Let V denote a set of N vertices. To construct a hypergraph process, create a new hyperedge at each event time of a Poisson process; the cardinality K of this hyperedge is random, with generating… Expand