Non-equilibrium Dynamics and Random Matrices

The (weakly) self-avoiding walk is a basic model of paths on the d-dimensional integer lattice that do not intersect (have few intersections), of interest from several different perspectives. I will discuss a proof that, in dimension 4, the susceptibility of the weakly self-avoiding walk diverges with an explicit logarithmic correction as the critical point is approached. The argument is based on a representation of the weakly self-avoiding walk as a supersymmetric field theory which is studied with a renormalization group method. This is joint work with Brydges and Slade.