Transition between Airy₁ and Airy₂ processes and TASEP fluctuations

Abstract

Abstract We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions, starting with certain deterministic initial conditions. For large time t , one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy 1 and Airy 2 processes, whose one‐point distributions are the GOE and GUE Tracy‐Widom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its one‐point distribution is a new interpolation between GOE and GUE edge distributions. © 2007 Wiley Periodicals, Inc.