2010/03/31 by Gideon Amir, Ivan Corwin, Jeremy Quastel · 3 citations
Mathematics · Physics and Astronomy · #math.PR #cond-mat.stat-mech #math-ph #math.MP #msc:82C22 #msc:60H15
paper · pdf · doi:10.1002/cpa.20347
published as Comm. Pure Appl. Math, 64: 466-537, 2011 · 68 pages, 2 figures. some typos and minor errata fixed
arxiv created 2010/09/27 · arxiv updated 2011/03/01
We consider the solution of the stochastic heat equation ∂T Z = 1/2 ∂X2 Z - Z \mathscrW with delta function initial condition Z (T=0)= δ0 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.