Suppose we immerse a large porous stone in a bucket of water. What is the probability that
the centre of the stone is wetted? In formulating a simple stochastic model for such a …

This course aims to be a (nearly) self-contained account of part of the mathematical theory of
percolation and related topics. The first nine chapters summarise rigorous results in …

These notes have been the subject of a course I gave in the summer 1997 for the school in
probability theory in Saint Flour. I review in a self {contained way the state of the art …

This self-contained book is a graduate-level introduction for mathematicians and for
physicists interested in the mathematical foundations of the field, and can be used as a …

Publisher Summary This chapter discusses the random geometry of equilibrium phases.
Percolation will come into play here on various levels. Its concepts like clusters, open paths …

The Gibbs point processes (GPP) constitute a large class of point processes with interaction
between the points. The interaction can be attractive, repulsive, depending on geometrical …

We analyze Markov chains for generating a random k‐coloring of a random graph Gn, d/n.
When the average degree d is constant, a random graph has maximum degree Θ (log n/log …

Suppose a new deck of cards is shuffled. Then as the number of shuffles grows, the
distribution of the deck becomes closer and closer to uniform over the set of permutations of …

We present a perfect simulation algorithm for measures that are absolutely continuous with
respect to some Poisson process and can be obtained as invariant measures of birth-and …

We present a new and simple approach to concentration inequalities in the context of
dependent random processes and random fields. Our method is based on coupling and …