Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension

We consider two models for directed polymers in space-time independent random media (the O'Connell-Yor semidiscrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time τ, the probability distributions for the free energy fluctuations, when rescaled by τ [superscript 1 over 3], converges to the GUE Tracy-Widom distribution. We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semidiscrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik–Ben Arous–Péché distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm—the Kardar-Parisi-Zhang equation. The Laplace transform formula we prove can be inverted to give the one-point probability distribution of the solution to these stochastic PDEs for the class of initial data.National Science Foundation (U.S.) (Grant DMS-1056390)National Science Foundation (U.S.) (Grant DMS-1208998)Clay Mathematics Institute (Clay Research Fellowship)Microsoft Research (Schramm Memorial Fellowship