Abstract

Understanding the behaviour of certain natural very long random curves in the plane is a seemingly simple question that has turned out to raise deep questions, some of which remain unsolved. For instance, theoretical physicists have predicted (and this is still an open problem) that the number a(N) of self-avoiding curves of length N on the square lattice Z × Z grows asymptotically like C ^N{11/32} for some constant C. More generally, theoretical physicists (Nienhuis, Cardy, Duplantier, Saleur etc.) have made predictions concerning the existence and values of critical exponents for various two-dimensional systems in statistical physics (such as self-avoiding walks, critical percolation, intersections of simple random walk) using considerations related to several branches of mathematics (probability theory, complex variables, representation theory of infinite-dimensional Lie algebras). We give a general introduction to the subject and briefly present some recent mathematical progress, including work of Kenyon, Werner, Smirnov, Lawler, and Schramm.