For many applications it is useful to sample from a nite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M suuciently large, the distribution… (More)
We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by… (More)
We prove that every bounded Lipschitz function F on a subset Y of a length space We also prove the first general uniqueness results for ∆ ∞ u = g on bounded subsets of R n (when g is uniformly continuous and bounded away from 0), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u ε (x) be the… (More)
We consider the random 2-satisfiability problem, in which each instance is a formula that is the conjunction of m clauses of the form x ∨ y, chosen uniformly at random from among all 2-clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n → α, the problem is known to have a phase transition at α c = 1, below which… (More)
Although research indicates that intervention programs can reduce overall recidivism rates among juvenile offenders , inadequate attention has been paid to their impact on serious juvenile offenders. This Bulletin describes a meta-analysis that addresses the following questions: Can intervention programs reduce re-cidivism rates among serious delin-quents?… (More)
We give a rigorous and self-contained survey of the abelian sand-pile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.
A random walk on Z d is excited if the first time it visits a vertex there is a bias in one direction, but on subsequent visits to that vertex the walker picks a neighbor uniformly at random. We show that excited random walk on Z d is transient iff d > 1. A random walk on Z d is excited (with bias ε/d) if the first time it visits a vertex it steps right… (More)