Free energy fluctuations for directed polymers in random media in 1+1 dimension

Understand

We consider two models for directed polymers in space-time independent random media (the O'Connell-Yor semi-discrete 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 tau, the probability distributions for the free energy fluctuations, when rescaled by tau^{1/3}, 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 semi-discrete 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-Peche distributions from spiked random matrix theory.