2010/06/30 by Ivan Corwin, Jeremy Quastel · 27 citations
Business, Management and Accounting · Mathematics · Physics and Astronomy · #Advanced Queuing Theory Analysis #Bernoulli's principle #Crossover #Dirac delta function #Distribution (mathematics) #Function (biology) #Limit (mathematics) #Moment (physics) #Random Matrices and Applications #Rarefaction (ecology) #Simple (philosophy) #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR
paper · pdf · doi:10.1214/11-aop725
published in The Annals of Probability 41(3A) (Institute of Mathematical Statistics) · Published in at http://dx.doi.org/10.1214/11-AOP725 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1003.0443
openalex publication_date 2013/04/29 · arxiv created 2013/05/24 · arxiv updated 2013/05/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with ρ-<ρ+ so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf–Cole solution of the Kardar–Parisi–Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [Comm. Pure Appl. Math. 64 (2011) 466–537] and [Nuclear Phys. B 834 (2010) 523–542]. At the edge of the rarefaction fan, the initial data is one-sided Brownian. We obtain a new family of crossover distributions giving the exact one-point distributions of this process, which converge, as T\nearrow∞ to those of the Airy A2→ BM process. As an application, we prove moment and large deviation estimates for the equilibrium Hopf–Cole solution of KPZ. These bounds rely on the apparently new observation that the FKG inequality holds for the stochastic heat equation. Finally, via a Feynman–Kac path integral, the KPZ equation also governs the free energy of the continuum directed polymer, and thus our formula may also be interpreted in those terms.