MARK KAC SEMINAR

The self-avoiding walk is a fundamental model in probability, combinatorics and statistical mechanics, for which many of the basic mathematical problems remain unsolved. Recent and ongoing progress for the 4-dimensional self-avoiding walk has been based on a renormalisation group analysis.

In the second and third talks in the series we describe an implementation of the renormalisation group approach. Features to be discussed include:

1. the finite-range decomposition of the covariance which leads to a multi-scale analysis and corresponding infinite-dimensional dynamical system,
2. the identification of the finitely many marginal and relevant directions in the dynamical system and resulting flow of coupling constants,
3. the proof that the irrelevant (contracting) directions in the dynamical system are in fact contracting,
4. the identification of the logarithmic correction to scaling for the susceptibility in dimension 4.

This is based on collaborations with Roland Bauerschmidt and David Brydges. Attendance at the October 4th lecture is not a prerequisite for the November 8th lectures, and prior knowledge of the renormalisation group will not be assumed.

The abstract above is a combined abstract for both lectures. The afternoon lecture will continue where the morning lecture broke off.