Mathematics – Probability
Scientific paper
2010-03-01
Comm. Pure Appl. Math, 64: 466-537, 2011
Mathematics
Probability
68 pages, 2 figures. some typos and minor errata fixed
Scientific paper
10.1002/cpa.20347
We consider the solution of the stochastic heat equation \partial_T \mathcal{Z} = 1/2 \partial_X^2 \mathcal{Z} - \mathcal{Z} \dot{\mathscr{W}} with delta function initial condition \mathcal{Z} (T=0)= \delta_0 whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions -- the {\it crossover distributions} -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes.
Amir Gideon
Corwin Ivan
Quastel Jeremy
No associations
LandOfFree
Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-662613