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Free energy fluctuations for directed polymers in random media in 1+1 dimension. (English) Zbl 1295.82035
Summary: 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 \(\tau\), 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 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.

MSC:
82D60 Statistical mechanics of polymers
60J65 Brownian motion
15B52 Random matrices (algebraic aspects)
35Q82 PDEs in connection with statistical mechanics
44A10 Laplace transform
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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