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- ItemLarge time asymptotics of growth models on space-like paths II: PNG and parallel TASEP(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Borodin, Alexei; Ferrari, Patrik; Sasamoto, TomohiroWe consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy$_1$ process. This result holds more generally for the observation points located along any space-like path in the space-time plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update.
- ItemTransition between Airy1 and Airy2 processes and TASEP fluctuations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Borodin, Alexei; Ferrari, Patrik; Sasamoto, TomohiroWe 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.