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

2005/12/01 by Herbert Spohn · 2 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Brownian motion #Condensed matter physics #Diffusion-limited aggregation #Directed percolation #Eigenvalues and eigenvectors #Fractal #Mathematical analysis #Mathematical physics #Mathematics #Percolation (cognitive psychology) #Percolation threshold #Physics #Quantum mechanics #Random Matrices and Applications #Random matrix #Renormalization group #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Type (biology) #Universality (dynamical systems) #cond-mat.stat-mech

paper · pdf · doi:10.1016/j.physa.2006.04.006

Lecture Notes: Fundamental Problems in Statistical Mechanics XI, Leuven, September 4 - 16, 2005

arxiv created 2005/12/01 · openalex publication_date 2006/05/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Three models from statistical physics can be analyzed by employing space-time determinantal processes: (1) crystal facets, in particular the statistical properties of the facet edge, and equivalently tilings of the plane, (2) one-dimensional growth processes in the Kardar-Parisi-Zhang universality class and directed last passage percolation, (3) random matrices, multi-matrix models, and Dyson's Brownian motion. We explain the method and survey results of physical interest.

Citations

• Random matrices and determinantal processes
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• A Brownian-Motion Model for the Eigenvalues of a Random Matrix
• Exact Scaling Functions for One-Dimensional Stationary KPZ Growth
• Fluctuations of the One-Dimensional Polynuclear Growth Model in Half-Space
• Queuing transitions in the asymmetric simple exclusion process
• Polynuclear Growth on a Flat Substrate and Edge Scaling of GOE Eigenvalues
• Determinantal random point fields
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Limiting distributions for a polynuclear growth model with external sources

Cited by

• Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
• Inhomogeneous field theory inside the arctic circle