Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions

We consider the solution of the stochastic heat equation ∂_T Z= 1/2 ∂_X²Z- ZẆ with delta function initial condition Z(T=0,X)= δ-{X=0} whose logarithm, with appropriate normalization, is the free energy of the continuum directed polymer, or the Hopf-Cole solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions, the 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 process with antishock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes.