Probability on Graphs

This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics to the very edge of contemporary research with the minimum of technical fuss. Topics covered include random walk percolation self-avoiding walk interacting particle systems uniform spanning tree random graphs as well as the Ising Potts and random-cluster models for ferromagnetism and the Lorentz model for motion in a random medium. Schramm-Löwner evolutions (SLE) arise in various contexts. The choice of topics is strongly motivated by modern applications and focuses on areas that merit further research. Special features include a simple account of Smirnov's proof of Cardy's formula for critical percolation and a fairly full account of the theory of influence and sharp-thresholds. Accessible to a wide audience of mathematicians and physicists this book can be used as a graduate course text. Each chapter ends with a range of exercises.