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Probability distribution of the free energy of the continuum directed random polymer in $$1 + 1$$ dimensions. (English) Zbl 1222.82070
The fully stochastic version of the Kardar-Parisi-Zhang equation used to model a stochastic growth of one-dimensional interface, while driven by the Gaussian space-time white noise, is still far from being a satisfactory theory, specifically since it is restricted to an equilibrium regime. The main goal of the present article is to address the far from equilibrium regime, which can be most conveniently stated in terms of a stochastic heat equation with delta function initial condition. The main result is an exact formula for the probability distribution for the free energy of the continuum directed random polymer in $$1+1$$ dimensions. Told otherwise one obtains the one-point distribution for the pertinent stochastic heat equation, or for the KPZ equation with narrow wedge initial conditions. Explicit formulas are obtained for the one-dimensional crossover (marginal) distributions which interpolate between standard Gaussian distribution for small times and the GUE Tracy-Widom one for large times. The proof heavily relies on a rigorous steepest descent analysis of the Tracy-Widom formula for the antisymmetric simple exclusion process with antishock initial data.

##### MSC:
 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics 82D60 Statistical mechanical studies of polymers 82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics 82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses) 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 15B52 Random matrices (algebraic aspects) 35Q53 KdV equations (Korteweg-de Vries equations)
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