Directed random polymers via nested contour integrals
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Borodi_Directed random.pdf
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Author(s)
Borodin, Alexei
Bufetov, Alexey
Corwin, Ivan
Date Issued
February 2016
Journal
Annals of Physics
Publisher
Elsevier
Citation
Borodin, Alexei et al. “Directed Random Polymers via Nested Contour Integrals.” Annals of Physics 368 (May 2016): 191–247 © 2016 Elsevier Inc
Version
Original manuscript
Abstract
We study the partition function of two versions of the continuum directed polymer in 1 + 1 dimension. In the full-space version, the polymer starts at the origin and is free to move transversally in R, and in the half-space version, the polymer starts at the origin but is reflected at the origin and stays in R_. The partition functions solve the stochastic heat equation in full-space or half-space with mixed boundary condition at the origin; or equivalently the free energy satisfies the Kardar-Parisi-Zhang equation.We derive exact formulas for the Laplace transforms of the partition functions. In the full-space this is expressed as a Fredholm determinant while in the half-space this is expressed as a Fredholm Pfaffian. Taking long-time asymptotics we show that the limiting free energy fluctuations scale with exponent 1/3 and are given by the GUE and GSE Tracy-Widom distributions. These formulas come from summing divergent moment generating functions, hence are not mathematically justified.The primary purpose of this work is to present a mathematical perspective on the polymer replica method which is used to derive these results. In contrast to other replica method work, we do not appeal directly to the Bethe ansatz for the Lieb-Liniger model but rather utilize nested contour integral formulas for moments as well as their residue expansions. Keywords: Kardar–Parisi–Zhang; Directed polymers; Bethe ansatz; Lieb–Liniger model; Delta Bose gas
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
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DOI of Published Version
https://doi.org/10.1016/J.AOP.2016.02.001
