Macro and Microscopic Limits of Directed Polymer Models

Abstract

Directed polymer models are well known Gibbs measures on random walk paths. Canonically they are defined so as to tilt the path distribution towards regions of space-time where an independent random field happens to be large, and as a result the paths tend to exhibit superdiffusive Kardar-Parisi-Zhang type fluctuation exponents, somehow betraying their random walk upbringing. Constructing these models on in the discrete space-time setting with a finite time horizon is straightforward, but extending them to infinite time horizons is difficult even in the fully discrete setting. I will review some relatively recent progress in the discrete and semi-discrete setting by myself and several other authors, some previous work of myself, Khanin, and Quastel on constructing continuous space-time models in the finite time horizon setting, and some attempts in progress to connect the two.