Self Avoiding Walk and Differential Forms

The continuous time simple random walk on a finite lattice will be reviewed. Any such walk defines a collection of local times \tau_{x} - the times spent at lattice sites x. There is an isomorphism relating these local times to finite dimensional integrals involving differential forms \phi \bar{\phi} + d\phi d\bar{\phi}. This is called the ``supersymmetric representation'' in physics. In principle the self-avoiding walk problem can be solved by approximately evaluating such integrals, provided the approximation is uniform in the dimension of the integral. The physicists have developed a far reaching extension of "Laplaces's method" which appears to be uniform in dimension. Formulas for coefficients in this approximation lead to the famous Feynman graphs.

Refreshments will be served in Math Annex Room 1115, 3:15 p.m.