Understanding the asymptotic behaviour when N goes to infinity of the number a(N) of self-avoiding curves of length N on the square lattice Z x Z is an tantilising open problem. Theoretical physicists (Nienhuis, Cardy, Duplantier etc.) have given various predictions concerning the existence and value of critical exponents for some two-dimensional systems in statistical physics (such as self-avoiding walks, critical percoaltion, intersections of simple random walk) using considerations related to several branches of mathematics (probability theory, complex variables, representation theory of infinite-dimensional Lie algebras). For instance, it is expected that a(N) grows asymptotically like cN N11/32 for some lattice-dependent constant c.
The aim of this lecture is to give an introduction to the subject and to present some mathematical progress obtained in joint work with Greg Lawler and Oded Schramm. In particular, we will briefly describe the proofs of the conjectures concerning intersections of simple random walks and Brownian motions and of Mandelbrot's conjecture on the fractal dimension of the outer frontier of a planar Brownian path ("its Hausdorff dimension is 4/3").
Part of this talk will be based on the review paper
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