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Author: Borodin, Alexei ×

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    Anisotropic growth of random surfaces in 2 + 1 dimensions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Borodin, Alexei; Ferrari, Patrik L.
    We construct a family of stochastic growth models in $2+1$ dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield $1+1$ dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models. The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece. (2) The one-point fluctuations of the height function in the curved part are asymptotically normal with variance of order $ln(t)$ for time $tgg 1$. (3) There is a map of the $(2+1)$-dimensional space-time to the upper half-plane $H$ such that on space-like submanifolds the multi-point fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on $H$.
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    Large time asymptotics of growth models on space-like paths I: PushASEP
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Borodin, Alexei; Ferrari, Patrik
    We 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.
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    Large 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, Tomohiro
    We 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.
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    Transition between Airy1 and Airy2 processes and TASEP fluctuations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Borodin, Alexei; Ferrari, Patrik; Sasamoto, Tomohiro
    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.