Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals

  title={Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals},
  author={Herbert Spohn},
  journal={Physica A-statistical Mechanics and Its Applications},
  • H. Spohn
  • Published 1 December 2005
  • Physics
  • Physica A-statistical Mechanics and Its Applications
From interacting particle systems to random matrices
In this contribution we consider stochastic growth models in the Kardar–Parisi–Zhang universality class in 1 + 1 dimension. We discuss the large time distribution and processes and their dependence
A combinatorial identity for the speed of growth in an anisotropic KPZ model
The speed of growth for a particular stochastic growth model introduced by Borodin and Ferrari in [Comm. Math. Phys. 325 (2014), 603-684], which belongs to the KPZ anisotropic universality class, was
Extreme value problems in random matrix theory and other disordered systems
We review some applications of central limit theorems and extreme values statistics in the context of disordered systems. We discuss several problems, in particular concerning random matrix theory
From interacting particle systems to random matrices Contribution to StatPhys 24 special issue
In this contribution we consider stochastic growth models in the Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large time distribution and processes and their dependence on
The Kardar-Parisi-Zhang equation - a statistical physics perspective
The article covers the one-dimensional Kardar-Parisi-Zhang equation, weak drive limit, universality, directed polymers in a random medium, replica solutions, statistical mechanics of line ensembles,
Symmetrized Models of Last Passage Percolation and Non-Intersecting Lattice Paths
A theory of the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups U(l), Sp(2l) and O(l) based on non-intersecting lattice paths, and integration techniques familiar from the theory ofrandom matrices.
Brownian Bridges for Late Time Asymptotics of KPZ Fluctuations in Finite Volume
Height fluctuations are studied in the one-dimensional totally asymmetric simple exclusion process with periodic boundaries, with a focus on how late time relaxation towards the non-equilibrium
Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
The studies of fluctuations of the one-dimensional Kardar–Parisi–Zhang universality class using the techniques from random matrix theory are reviewed from the point of view of the asymmetric simple
Fluctuation Properties of the TASEP with Periodic Initial Configuration
Abstract We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a
The Kardar-Parisi-Zhang equation and universality class
Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or


One-dimensional stochastic growth and Gaussian ensembles of random matrices
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
Universal distributions for growth processes in 1+1 dimensions and random matrices
A scaling theory for Kardar-Parisi-Zhang growth in one dimension is developed by a detailed study of the polynuclear growth model and three universal distributions for shape fluctuations and their dependence on the macroscopic shape are identified.
Stochastic Surface Growth
Growth phenomena constitute an important field in nonequilibrium statistical mechanics. Kardar, Parisi, and Zhang (KPZ) in 1986 proposed a continuum theory for local stochastic growth predicting
21pYO-3 Spatial correlations of the 1D KPZ surface on a flat substrate
We study the spatial correlations of the one-dimensional KPZ surface for the flat initial condition. It is shown that the multi-point joint distribution for the height is given by a Fredholm
Statistical Self-Similarity of One-Dimensional Growth Processes
Exact Scaling Functions for One-Dimensional Stationary KPZ Growth
We determine the stationary two-point correlation function of the one-dimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence
Non-intersecting paths, random tilings and random matrices
Abstract. We investigate certain measures induced by families of non-intersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abc-hexagon, a dimer model on a cylindrical brick
Polynuclear Growth on a Flat Substrate and Edge Scaling of GOE Eigenvalues
We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the
Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their
Fluctuations of the One-Dimensional Polynuclear Growth Model in Half-Space
We consider the multi-point equal time height fluctuations of the one-dimensional polynuclear growth model in half-space. For special values of the nucleation rate at the origin, the multi-layer