Self-Interacting Walk and Functional Integration

These lectures are directed at analysts who are interested in learning some of the standard tools of theoretical physics, including functional integrals, the Feynman expansion, supersymmetry and the Renormalization Group. These lectures are centered on the problem of determining the asymptotics of the end-to-end distance of a self-avoiding walk on a $D$-dimensional simple cubic lattice as the number of steps grows. When $D=4$ the end-to-end distance has been conjectured to grow as Const. $n^{1/2}\log^{1/8}n,$ where $n$ is the number of steps. We include a theorem, obtained in joint work with John Imbrie, that validates the $D=4$ conjecture in the simplified setting known as the ``Hierarchial Lattice.''