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Low-temperature localization of directed polymers

Seminar: Probability Seminar | February 21 | 3:10-4 p.m. | 1011 Evans Hall

Erik Bates, Stanford University

Department of Statistics

On the d-dimensional integer lattice, directed polymers are paths of a random walk that have been reweighted according to a random environment that refreshes at each time step. The qualitative behavior of the system is governed by a temperature parameter; if this parameter is small, the environment has little effect, meaning all possible paths are close to equally likely. If the parameter is made large, however, the system undergoes a phase transition at which the path’s endpoint starts to localize. To understand the extent of this localization, we exploit the underlying Markov structure of the quenched endpoint distribution. 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. (joint work with Sourav Chatterjee)