期刊
COMMUNICATIONS IN MATHEMATICAL PHYSICS
卷 324, 期 1, 页码 215-232出版社
SPRINGER
DOI: 10.1007/s00220-013-1750-x
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资金
- NSF [DMS-1056390, DMS-1208998]
- Clay Research Fellowship
- Microsoft Research through the Schramm Memorial Fellowship
- Natural Science and Engineering Research Council of Canada
- Fields-Ontario Postdoctoral Fellowship
- Fondecyt [1120309]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1438867] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1056390] Funding Source: National Science Foundation
We prove that under n (1/3) scaling, the limiting distribution as n -> infinity 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.
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