Call a disjunction of k Boolean variables a k-clause. For a set V of n Boolean variables, let Ck(V ) denote the set of all 2n possible k-clauses on V . A random k-CNF formula Fk(n,m) is formed byâ€¦ (More)

Abstract: We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and inâ€¦ (More)

It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problemsâ€¦ (More)

A <i>local graph partitioning algorithm</i> finds a set of vertices with small conductance (i.e.~a sparse cut) by adaptively exploring a large graph G, starting from a specified vertex. For theâ€¦ (More)

We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF)â€¦ (More)

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguingâ€¦ (More)

We study continuous 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. Weâ€¦ (More)

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric propertiesâ€¦ (More)

2 1 Preface These notes are based on Lectures delivered at the Saint Flour Summer School in July 1997. Accurate notes were taken by Dimitris Gatzouras, who also edited them. Revisions by David Levinâ€¦ (More)