Log-Gamma Polymer Free Energy Fluctuations via a Fredholm Determinant Identity
Author(s)Borodin, Alexei; Corwin, Ivan; Remenik, Daniel
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We prove that under n[superscript 1/3] scaling, the limiting distribution as n → ∞ of the free energy of Seppalainen’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 of Macdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach.
Original manuscript June 20, 2012
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Communications in Mathematical Physics
Borodin, Alexei, Ivan Corwin, and Daniel Remenik. “Log-Gamma Polymer Free Energy Fluctuations via a Fredholm Determinant Identity.” Communications in Mathematical Physics (July 3, 2013).