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- ItemThe Airy1 process is not the limit of the largest eigenvalue in GOE matrix diffusion(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Bornemann, Folkmar; Ferrari, Patrik; Prähofer, MichaelUsing a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1-process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion
- ItemThe universal Airy1 and Airy2 processes in the totally asymmetric simple exclusion process(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Ferrari, PatrikIn the totally asymmetric simple exclusion process (TASEP) two processes arise in the large time limit: the Airy1 and Airy2 processes. The Airy2 process is an universal limit process occurring also in other models: in a stochastic growth model on 1 + 1-dimensions, 2d last passage percolation, equilibrium crystals, and in random matrix diffusion. The Airy1 and Airy2 processes are defined and discussed in the context of the TASEP. We also explain a geometric representation of the TASEP from which the connection to growth models and directed last passage percolation is immediate.
- ItemLarge time asymptotics of growth models on space-like paths I: PushASEP(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Borodin, Alexei; Ferrari, PatrikWe consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any space-like path are described by the Airy$_1$ and Airy$_2$ processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.
- 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.
- ItemSlow decorrelations in KPZ growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ferrari, PatrikFor stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class in $1+1$ dimensions, fluctuations grow as $t^1/3$ during time $t$ and the correlation length at a fixed time scales as $t^2/3$. In this note we discuss the scale of time correlations. For a representant of the KPZ class, the polynuclear growth model, we show that the space-time is non-trivially fibred, having slow directions with decorrelation exponent equal to $1$ instead of the usual $2/3$. These directions are the characteristic curves of the PDE associated to the surface's slope. As a consequence, previously proven results for space-like paths will hold in the whole space-time except along the slow curves.