O ct 2 00 2 Conformal Invariance in Percolation , Self - Avoiding Walks , and Related Problems ∗

Over the years, problems like percolation and self-avoiding walks have provided important testing grounds for our understanding of the nature of the critical state. I describe some very recent ideas, as well as some older ones, which cast light both on these problems themselves and on the quantum field theories to which they correspond. These ideas come from conformal field theory, Coulomb gas mappings, and stochastic Loewner evolution. This talk is about ‘geometric’ critical phenomena. These are random spatial processes, where either (1) the probability distribution is determined by equilibrium statistical mechanics, and we ask questions about geometrical properties, or (2) the probability distribution is itself geometrical in nature. The simplest example of (1) is clustering in percolation (see Fig. 1), in which the probability distribution ∗Plenary talk given at the International Conference on Theoretical Physics, Paris, July 2002.