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Based on a lecture course given at Chalmers University of Technology, this 2002 book is ideal for advanced undergraduate or beginning graduate students. The author first develops the necessary background in probability theory and Markov chains before applying it to study a range of randomized algorithms with important applications in optimization and other problems in computing. Amongst the algorithms covered are the Markov chain Monte Carlo method, simulated annealing, and the recent Propp-Wilson algorithm. This book will appeal not only to mathematicians, but also to students of statistics and computer science. The subject matter is introduced in a clear and concise fashion and the numerous exercises included will help students to deepen their understanding.

We establish phase transitions for a class of continuum multi-type particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of Lebowitz and Lieb. A phase transition still occurs when we allow a background pair interaction (between all particles) which is superstable and has sufficiently short range of repulsion. Our approach involves a random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. In the course of our argument, we establish the existence of a percolation transition for Gibbsian particle systems with random edges between the particles, and also give an alternative proof for the existence of Gibbs measures with superstable interaction.

Place a water glass at each integer point, the one at the origin being full and all others empty, and consider averaging procedures where we repeatedly pick a pair of adjacent glasses and pool their contents, leaving the two glasses with equal amounts but with the total amount unchanged. Some simple results are derived for what kinds of configurations of water levels are obtainable via such procedures. These are applied in the analysis of the so-called Deffuant model for social interaction, where individuals have opinions represented by numbers between 0 and 1, and whenever two individuals interact they take a step towards equalizing their opinion, unless their opinions differ beyond a fixed amount θ in which case they make no adjustment. In particular, we reprove and sharpen the recent result of Lanchier which identifies the critical value θ c for consensus formation in the Deffuant model on Z to be 1 2 .

An interacting particle system modelling competing growth on the Z 2 lattice is defined as follows. Each x ∈ Z 2 is in one of the states {0, 1, 2}. 1's and 2's remain in their states forever, while a 0 flips to a 1 (resp. a 2) at a rate equal to the number of its neighbours which are in state 1 (resp. 2). This is a generalization of the well known Richardson model. 1's and 2's may be thought of as two types of infection, and 0's as uninfected sites. We prove that if we start with a single site in state 1 and a single site in state 2, then there is positive probability for the event both types of infection reach infinitely many sites. This result implies that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.