In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stochastic processes hardly touches on the sort of non-asymptotic analysis required in this application. As a consequence, it had previously not been possible to make useful, mathematically rigorous statements about the quality of the estimates obtained. Within the last ten years, analytical tools have been devised with the aim of correcting this deficiency. As well as permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics, the introduction of these tools has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization. The “Markov chain Monte Carlo ” method has been applied to a variety of such problems, and often provides the only known efficient (i.e., polynomial time) solution technique.

CONTENTS 0 PREFACE 3 1 BASICS OF PROBABILITY THEORY 5 2 MARKOV CHAINS 12 3 COMPUTER SIMULATION OF MARKOV CHAINS 19 4 IRREDUCIBLE AND APERIODIC MARKOV CHAINS 25 5 STATIONARY DISTRIBUTIONS 30 6 REVERSIBLE MARKOV CHAINS 41 7 MARKOV CHAIN MONTE CARLO 46 8 FAST CONVERGENCE OF MCMC ALGORITHMS 54 9 APPROXIMATE COUNTING 63 10 THE PROPP--WILSON ALGORITHM 74 11 PROPP--WILSON WITH READ-ONCE RANDOMNESS 82 12 SIMULATED ANNEALING 88 13 FURTHER READING 96 1 2 0 PREFACE The first version of these lecture notes was composed for a new course on randomized algorithms at Chalmers University of Technology, in the spring semester 2000. The amount of material contained therein was not quite enough for an entire course, and the idea was to use these notes together with the book "Randomized Algorithms" by Motwani & Raghavan [MR]. The current version of 2001 has now, with the addition of three more chapters (Chapters 8, 9 and 11), expanded into something more "course-sized", bu

Random independent sets in graphs arise, for example, in statistical physics, in the hard-core model of a gas. A new rapidly mixing Markov chain for independent sets is defined in this paper. We show that it is rapidly mixing for a wider range of values of the parameter than the Luby-Vigoda chain, the best previously known. Moreover the new chain is apparently more rapidly mixing than the Luby-Vigoda chain for larger values of (unless the maximum degree of the graph is 4). An extension of the chain to independent sets in hypergraphs is described. This chain gives an efficient method for approximately counting the number of independent sets of hypergraphs with maximum degree two, or with maximum degree three and maximum edge size three. Finally, we describe a method which allows one, under certain circumstances, to deduce the rapid mixing of one Markov chain from the rapid mixing of another, with the same state space and stationary distribution. This method is applied to two Markov ch...

A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or "Glauber dynamics." Typically these single site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding non-local algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of c...

We define a new Markov chain on (proper) k-colourings of graphs, and relate its convergence properties to the maximum degree \Delta of the graph. The chain is shown to have bounds on convergence time appreciably better than those for the wellknown Jerrum/Salas--Sokal chain in most circumstances. For the case k = 2\Delta, we provide a dramatic decrease in running time. We also show improvements whenever the graph is regular, or fewer than 3\Delta colours are used. The results are established using the method of path coupling. We indicate that our analysis is tight by showing that the couplings used are optimal in a sense which we define. 1 Introduction Markov chains on the set of proper colourings of graphs have been studied in computer science [9] and statistical physics [13]. In both applications, the rapidity of convergence of the chain is the main focus of interest, though for somewhat different reasons. The papers [9, 13] introduced a simple Markov chain, which we shall refer to a...

The Swendsen-Wang process provides one possible dynamics for the Q-state Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The legitimacy of such simulations depends on the rate of convergence of the process to equilibrium, often known as the mixing rate. Empirical observations suggest that the Swendsen-Wang process mixes rapidly in many instances of practical interest. In spite of this, we show that there are occasions on which the Swendsen-Wang process requires exponential time (in the size of the system) to approach equilibrium.

We consider the problem of generating a random q-colouring of a graph G = (V, E).

7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that are hard to compute exactly. The quantity z of interest is expressed as the expectation z = ExpZ of a random variable (r.v.) Z for which some efficient sampling procedure is available. By taking the mean of some sufficiently large set of independent samples of Z, one may obtain an approximation to z. For example, suppose S = \Phi (x; y) 2 [0; 1] 2 : p i (x; y) 0; for all i \Psi<F12

We present two algorithms for uniformly sampling from the proper colorings of a graph using k colors. We use exact sampling from the stationary distribution of a Markov chain with states that are the k-colorings of a graph with maximum degree ¢. As opposed to approximate sampling algorithms based on rapid mixing, these algorithms have termination criteria that allow them to stop on some inputs much more quickly than in the worst case running time bound. For the first algorithm we show that when ¡¤£, the algorithm has an upper limit on the expected running time that is polynomial. For the second algorithm we show that for ¡�£�� where ¢� � � is an integer that satisfies ����£�� the running time is polynomial. Previously, Jerrum showed that it was possible to approximately sample uniformly in polynomial time from the set of ¡-colorings when ¡�� but our algorithm is the first polynomial time exact sampling algorithm for this problem. Using approximate sampling, Jerrum also showed how to approximately count the number of ¡-colorings. We give a new procedure for approximately counting the number of ¡-colorings that improves the running time of the procedure of Jerrum by a factor of is the number of nodes in �� � when ¥����, where �� ¢ � the graph to be colored � and is the number of edges. In addition, we present an improved analysis of the chain of Luby and Vigoda for exact sampling from the independent sets of a graph. Finally, we present the first polynomial time method for exactly sampling from the sink free orientations of a graph. Bubley and Dyer showed how to approximately sample from this state space ��¥���������¥��������� � in time, our algorithm ��¥��¦�� � takes expected time.

We study discrete time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap 1 \Gamma 2 ) for the dynamics on trees and on certain hyperbolic graphs, is polynomial in n. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that if the relaxation time 2 satisfies 2 = O(n), then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp. 1.