Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions
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Publication:3168879
DOI10.1002/CPA.20347zbMATH Open1222.82070arXiv1003.0443OpenAlexW2149433831MaRDI QIDQ3168879
Author name not available (Why is that?)
Publication date: 27 April 2011
Published in: (Search for Journal in Brave)
Abstract: We consider the solution of the stochastic heat equation partial_T mathcal{Z} = 1/2 partial_X^2 mathcal{Z} - mathcal{Z} dot{mathscr{W}} with delta function initial condition mathcal{Z} (T=0)= delta_0 whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions -- the {it crossover distributions} -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes.
Full work available at URL: https://arxiv.org/abs/1003.0443
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