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. 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.Comment: 73 pages, 10 figure
