Polymers in Probability: Bridges, Brownian Motion, and Disorder on an Intermediate Scale

Job Candidate Talk
Tuesday, December 11, 2012 - 11:05
1 hour (actually 50 minutes)
Skiles 005
Chemical polymers are long chains of molecules built up from many individual monomers. Examples are plastics (like polyester and PVC), biopolymers (like cellulose, DNA, and starch) and rubber. By some estimates over 60% of research in the chemical industry is related to polymers. The complex shapes and seemingly random dynamics inherent in polymer chains make them natural candidates for mathematical modelling. The probability and statistical physics literature abounds with polymer models, and while most are simple to understand they are notoriously difficult to analyze.      In this talk I will describe the general flavor of polymer models and then speak more in depth on my own recent results for two specific models. The first is the self-avoiding walk in two dimensions, which has recently become amenable to study thanks to the invention of the Schramm-Loewner Evolution. Joint work with Hugo-Duminil Copin shows that a specific feature of the self-avoiding walk, called the bridge decomposition, carries over to its conjectured scaling limit, the SLE(8/3) process. The second model is for directed polymers in dimension 1+1. Recent joint work with Kostya Khanin and Jeremy Quastel shows that this model can be fully understood when one considers the polymer in the previously undetected "intermediate" disorder regime. This work ultimately leads to the construction of a new type of diffusion process, similar to but actually very different from Brownian motion.