## 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.

##### Description

Original manuscript June 20, 2012

##### Date issued

2013-07##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Communications in Mathematical Physics

##### Publisher

Springer-Verlag

##### Citation

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).

Version: Original manuscript

##### ISSN

0010-3616

1432-0916