Differential Equations for Dyson Processes

Abstract

We call "Dyson process" any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n by n Hermitian matrices, and the eigenvalues describe n curves. Given sets X_1,...,X_m the probability that for each k no curve passes through X_k at time τ_k is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the Laguerre process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite process at the edge leads to the Airy process and in the bulk to the sine process; scaling the Laguerre process at the edge leads to the Bessel process. Generalizing and strengthening earlier work, we assume that each X_k is a finite union of intervals and find for the Airy process a system of partial differential equations, with the end-points of the intervals of the X_k as independent variables, whose solution determines the probability that for each k no curve passes through X_k at time τ_k. Then we find the analogous systems for the Hermite process (which is more complicated) and also for the sine process. Finally we find an analogous system of PDEs for the Bessel process, which is the most difficult.

36 pages, LaTeX. Version 3 corrects an error in the earlier versions

References in corpus (9)

• Level-Spacing Distributions and the Airy Kernel
• Discrete polynuclear growth and determinantal processes
• Fredholm Determinants, Differential Equations and Matrix Models
• Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices
• Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
• On some integrals over the U(N) unitary group and their large N limit
• Toeplitz determinants, random growth and determinantal processes
• A PDE for the joint distributions of the Airy Process
• Universal parametric correlations in the transmission eigenvalue spectra of disordered conductors

Cited by in corpus (50)

• Eynard-Mehta theorem, Schur process, and their pfaffian analogs
• Spatial correlations of the 1D KPZ surface on a flat substrate
• The Pearcey Process
• Airy kernel with two sets of parameters in directed percolation and random matrix theory
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• Fluctuations of a one-dimensional polynuclear growth model in a half space
• Airy processes and variational problems
• Nonintersecting Brownian excursions
• Noncolliding Brownian Motion and Determinantal Processes
• Non-Equilibrium Dynamics of Dyson's Model with an Infinite Number of Particles
• Noncolliding Squared Bessel Processes
• Airy processes with wanderers and new universality classes
• Nonintersecting Brownian motions on the unit circle
• Infinite systems of non-colliding generalized meanders and Riemann-Liouville differintegrals
• Universality of the Pearcey process
• One-dimensional stochastic growth and Gaussian ensembles of random matrices
• Riemann-Hilbert approach to multi-time processes; the Airy and the Pearcey case
• Markov property of determinantal processes with extended sine, Airy, and Bessel kernels
• Determinantal Martingales and Noncolliding Diffusion Processes
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• Noncolliding processes, matrix-valued processes and determinantal processes
• From the Pearcey to the Airy process
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• The partition function of the two-matrix model as an isomonodromic tau-function
• Multiplicative functionals on ensembles of non-intersecting paths
• Nonlinear PDEs for gap probabilities in random matrices and KP theory
• Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
• Markov processes on partitions
• On the partial connection between random matrices and interacting particle systems
• The hard-edge tacnode process for Brownian motion
• Determinantal process starting from an orthogonal symmetry is a Pfaffian process
• Average characteristic polynomials in the two-matrix model
• Asymptotics for the Covariance of the Airy_2 process
• The correspondence between Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches in random matrix theory: the Gaussian case
• The Airy${}_2$ process and the 3D Ising model
• Product matrix processes for coupled multi-matrix models and their hard edge scaling limits
• Dyson's non-intersecting Brownian motions with a few outliers
• Random skew plane partitions and the Pearcey process
• Non-colliding Jacobi processes as limits of Markov chains on Gelfand-Tsetlin graph
• Exact solution of interacting particle systems related to random matrices
• Stochastic Growth in One Dimension and Gaussian Multi-Matrix Models
• Correlation function of the Schur process with a fixed final partition
• Characteristic Polynomials of Random Matrices and Noncolliding Diffusion Processes
• Aspects géométriques et intégrables des modèles de matrices aléatoires
• Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles
• Various equations for gap probabilities of coupled Gaussian matrices
• On the eigenvalue process of a matrix fractional Brownian motion
• Determinantal Point Processes, Stochastic Log-Gases, and Beyond
• Determinantal identity for multilevel systems and finite determinantal processes
• Determinantal Martingales and Interacting Particle Systems