WIAS Preprint No. 1214, (2007)

Transition between Airy$_1$ and Airy$_2$ processes and TASEP fluctuations



Authors

  • Borodin, Alexei
  • Ferrari, Patrik
  • Sasamoto, Tomohiro

2010 Mathematics Subject Classification

  • 82C22 60K35 15A52

Keywords

  • Simple exclusion process, universality, KPZ class, Airy process, random matrices

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. F or 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 interpolati on between GOE and GUE edge distributions.

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