The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability and statistics. Some new results are given ae an illustration of the geometric theory of Markov chains.
We study the cutoff phenomenon for generalized riffle shuffles where, at each step, the deck of cards is cut into a random number of packs of multinomial sizes which are then riffled together.
We consider weighted graphs satisfying sub-Gaussian estimate for the natural random walk. On such graphs, we study symmetric Markov chains with heavy tailed jumps. We establish a threshold behavior of such Markov chains when the index governing the tail heaviness (or jump index) equals the escape time exponent (or walk dimension) of the sub-Gaussian… (More)
Let (M, d, µ) be a uniformly discrete metric measure space satisfying space homogeneous volume doubling condition. We consider discrete time Markov chains on M symmetric with respect to µ and whose one-step transition density is comparable to (V h (d(x, y))φ(d(x, y)) −1 , where φ is a positive continuous regularly varying function with index β ∈ (0, 2) and… (More)
We study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition. For any locally compact unimodular group G and any positive function : G → [0, +∞], we introduce a function G, which describes the fastest possible decay of n → φ (2n) (e) when φ is a symmetric continuous probability… (More)
We develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincaré inequality and reverse volume doubling property. We… (More)
The study of convolution powers of a finitely supported probability distribution φ on the d-dimensional square lattice is central to random walk theory. For instance, the nth convolution power φ (n) is the distribution of the nth step of the associated random walk and is described by the classical local limit theorem. Following previous work of P. Diaconis… (More)
Let G be a finitely generated group of polynomial volume growth equipped with a word-length | · |. The goal of this paper is to develop techniques to study the behavior of random walks driven by symmetric measures µ such that, for any ǫ > 0, | · | ǫ µ = ∞. In particular, we provide a sharp lower bound for the return probability in the case when µ has a… (More)