Title: Low-temperature localization of directed polymers

Date: 09/07/2017

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

On the d-dimensional integer lattice, directed polymers can be seen as paths of a random walk in random environment, except that the environment updates at each time step. The result is a statistical mechanical system, whose qualitative behavior is governed by a temperature parameter and the law of the environment. Historically, the phase transitions of this system have been best understood by whether or not the path’s endpoint localizes. While the endpoint is no longer a Markov process as in a random walk, its quenched distribution is. The key difficulty is that the space of measures is too large for one to expect convergence results. By adapting methods recently used by Mukherjee and Varadhan, we develop a compactification theory to resolve the issue. In this talk, we will discuss this intriguing abstraction, as well as new concrete theorems it allows us to prove for directed polymers constructed from SRW or any other walk. (This talk is based on joint work with Sourav Chatterjee.)