On the partial connection between random matrices and interacting particle systems

Abstract

In the last decade there has been increasing interest in the fields of random matrices, interacting particle systems, stochastic growth models, and the connections between these areas. For instance, several objects appearing in the limit of large matrices arise also in the long time limit for interacting particles and growth models. Examples of these are the famous Tracy-Widom distribution functions and the Airy_2 process. The link is however sometimes fragile. For example, the connection between the eigenvalues in the Gaussian Orthogonal Ensembles (GOE) and growth on a flat substrate is restricted to one-point distribution, and the connection breaks down if we consider the joint distributions. In this paper we first discuss known relations between random matrices and the asymmetric exclusion process (and a 2+1 dimensional extension). Then, we show that the correlation functions of the eigenvalues of the matrix minors for beta=2 Dyson's Brownian motion have, when restricted to increasing times and decreasing matrix dimensions, the same correlation kernel as in the 2+1 dimensional interacting particle system under diffusion scaling limit. Finally, we analyze the analogous question for a diffusion on (complex) sample covariance matrices.

31 pages, LaTeX; Added a section concerning the Markov property on space-like paths

References in corpus (18)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• On Orthogonal and Symplectic Matrix Ensembles
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Discrete polynuclear growth and determinantal processes
• Asymptotics in ASEP with Step Initial Condition
• Fluctuation properties of the TASEP with periodic initial configuration
• Eynard-Mehta theorem, Schur process, and their pfaffian analogs
• Spatial correlations of the 1D KPZ surface on a flat substrate
• Matrices coupled in a chain. I. Eigenvalue correlations
• Transition between Airy_1 and Airy_2 processes and TASEP fluctuations
• Finite volume partition functions and Itzykson-Zuber integrals
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• Differential Equations for Dyson Processes
• On some integrals over the U(N) unitary group and their large N limit
• The Airy_1 process is not the limit of the largest eigenvalue in GOE matrix diffusion
• Two speed TASEP
• Orbit measures, random matrix theory and interlaced determinantal processes

Cited by in corpus (6)

• Anisotropic growth of random surfaces in 2+1 dimensions
• Interlacing Diffusions
• A limit theorem at the spectral edge for corners of time-dependent Wigner matrices
• Cycles and eigenvalues of sequentially growing random regular graphs
• Perturbed GUE Minor Process and Warren's Process with Drifts
• Exact solution of interacting particle systems related to random matrices