Log-Gamma Polymer Free Energy Fluctuations via a Fredholm Determinant Identity

Abstract

We prove that under n1/3 scaling, the limiting distribution as n → ∞of the free energy of Seppäläinen’s log-Gamma discrete directed polymer is GUE Tracy- Widom. The main technical innovation we provide is a general identity between a class of n-fold contour integrals and a class of Fredholm determinants. Applying this identity to the integral formula proved in Corwin et al. (Tropical combinatorics and Whittaker functions. http://arxiv.org/abs/1110.3489v3 [math.PR], 2012) for the Laplace transform of the log-Gamma polymer partition function, we arrive at a Fredholm determinant which lends itself to asymptotic analysis (and thus yields the free energy limit theorem). The Fredholm determinant was anticipated in Borodin and Corwin (Macdonald processes. http://arxiv.org/abs/1111.4408v3 [math.PR], 2012) via the formalism ofMacdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach.