One-dimensional stochastic growth and Gaussian ensembles of random matrices

Abstract

In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of random matrices. This connection can be explained via point processes associated to the PNG model and the random matrices ensemble by an extension to the multilayer PNG and multi-matrix models, respectively. We also explain other models which are equivalent to the PNG model: directed polymers, the longest increasing subsequence problem, Young tableaux, a directed percolation model, kink-antikink gas, and Hammersley process.

34 pages, 8 figures; Review for the Inhomogeneous Random Systems 2005 proceedings; Final version

References in corpus (16)

• Discrete polynuclear growth and determinantal processes
• Exact scaling functions for one-dimensional stationary KPZ growth
• Eynard-Mehta theorem, Schur process, and their pfaffian analogs
• Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process
• Distribution functions for largest eigenvalues and their applications
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• Differential Equations for Dyson Processes
• Fluctuations of a one-dimensional polynuclear growth model in a half space
• A determinantal formula for the GOE Tracy-Widom distribution
• Exact Asymptotic Results for a Model of Sequence Alignment
• An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution
• Dynamical Correlations among Vicious Random Walkers
• On Asymptotics for the Airy Process
• Fluctuations of an Atomic Ledge Bordering a Crystalline Facet
• Polynuclear growth model, GOE${}^2$ and random matrix with deterministic source

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• Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP
• Slow decorrelations in KPZ growth
• Finite time corrections in KPZ growth models
• Universality for mathematical and physical systems
• Reunion probability of N vicious walkers: typical and large fluctuations for large N
• Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure
• Multiclass processes, dual points and M/M/1 queues
• Superdiffusivity of Asymmetric Energy Model in Dimension One and Two
• Hydrodynamic Limit of a (2+1)-Dimensional Crystal Growth Model in the Anisotropic KPZ Class
• Hydrodynamic limit for a 2D interlaced particle process