Seminar

Language English
From 2009/02/06 13:30
To 2009/02/06 15:00
PlaceRoom 552, Building No.3, Faculty of Science, Kyoto University
Seminar NameGCOE Lecture Series
Title A renormalisation group analysis of the 4-dimensional self-avoiding walk　
Field Analysis
SpeakersProf. Gordon Slade
AffiliationUniversity of British Columbia
AbstractSelf-avoiding walks on are simple-random walk paths without self-interstections. Self-avoiding walks of the same length are declared to be equally likely. Basic questions are: (1) how many self-avoiding walks are there of length (started from the origin), and (2) how far on average is their endpoint from the origin? The lace expansion has answered these questions in dimensions 5 and higher. For = 2, SLE appears to hold the key to the answer, but so far no one has understood how to unlock the door. For = 3, there are only numerical results.

In this mini-course, I will describe work in progress with David Brydges for the case = 4. Our immediate goal is to prove that the critical two-point function (Green function) for a spread-out model of self-avoiding walks on decays like at large distances, as it does for simple random walk.

We begin with an exact representation (due to John Imbrie) of the two-point function for self-avoiding walks as the two-point function of a certain field theory involving both bosons and fermions. In the first part of the course, I will explain this representation. Given the representation, we forget about the walks, and perform a renormalisation group analysis of the field theory. In the second part of the course, I will describe some of the ingredients in the renormalisation group analysis.

I will not assume that the audience has prior knowledge of field theory or the renormalisation group, and these concepts will be developed as the course proceeds.