Title: Duality and Supersymmetry in Statistical Mechanics and Random Matrices
Abstract: Random matrices appear in many parts of mathematics such as probability, statistics, number theory and the spectral theory of random Schrödinger operators. We show how certain models of supersymmetric spins provide a nonabelian dual representation for spectral problems in random matrix theory. Ordered and disordered phases correspond to different spectral types and time evolutions of a random matrix Hamiltonian. In 3 dimensions, a phase transition has been proved for a noncompact, supersymmetric statistical mechanics system. This model is equivalent to a classical history dependent walk which prefers to jump to lattice vertices it has visited in the past.