INVITATION for LECTURE SERIES by

 David Brydges, University of British Columbia, Canada
 

Friday June 13; 3-5.15 pm. (LG 1.105)
Fridays June 20, June 27, July 4 & July 11; 10-12.15 am (LG 1.105, on July 4 - LG 1.110)

* Please send an e-mail to coolen@eurandom.tue.nl if you plan to attend.

The lectures will cover

1. Models and their representation in terms of Gaussian integrals

2. Hierarchical models and the action of the renormalisation group

3. Renormalisation group for models on the Euclidean lattice

Abstract

In the theory of critical phenomena in statistical mechanics, the idea of a scaling limit is exemplified by observing a very long self-avoiding walk from far away so that individual steps become invisible and one sees (the ocupation density of) a path in the continuum. The scaling limit is the probability law for this random continuum path. The Renormalisation Group (RG) is a nascent program to construct and classify scaling limits. It is based on the Nobel prize work of Ken Wilson. RG is a map acting on a space of statistical mechanical models. Models are probability measures on random fields and RG acts on a model by integrating out the short distance fluctuations giving rise to a new model whose typical random field has the same long distance fluctuations but suppressed short distance fluctuations. Finding scaling limits corresponds to proving that the action of RG is almost equivalent to scaling. In order to apply our form of RG one must first express the problem as a “multiplicative perturbation” of the “massless Gaussian measure”. I will define these terms and show that a range of models can be put in this form, including finding the end-to-end distance of self-avoiding walk. Another simpler model presenting the same basic difficulties is “dipoles on the Euclidean lattice”. This is closely related to determining correlations for the “anharmonic bedspring” which is a model for sound waves in a crystal. I will use hierarchical models and then this dipole model as my main examples for detailed analysis.
My Park City Lecture notes posted here give an idea of what to expect.