Universality conjecture and results for a model of several coupled positive-definite matrices

Abstract

The paper contains two main parts: in the first part, we analyze the general case of $p\ge 2$ matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a $(p+1)\times (p+1)$ matrix valued solution of a Riemann-Hilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multi-level random determinantal point fields which include the Bessel${}_{\nu }$ universality for $1$-level and the Meijer-$G$ universality for $2$-level. Our analysis uses the Deift-Zhou nonlinear steepest descent method and the explicit construction of a $(p+1)\times (p+1)$ origin parametrix in terms of Meijer G-functions. The solution of the full Riemann-Hilbert problem is derived rigorously only for $p=3$ but the general framework of the proof can be extended to the Cauchy chain of arbitrary length $p$.

arXiv comment45 pages, 7 figures. To appear in Commun. Math. Phys. Version 2 contains minor changes in the introduction and updates literature

References in corpus (17)

• Level-Spacing Distributions and the Airy Kernel
• Level-Spacing Distributions and the Bessel Kernel
• Level-Spacing Distributions and the Airy Kernel
• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Matrices coupled in a chain. I. Eigenvalue correlations
• Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices
• Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits
• Singular value correlation functions for products of Wishart random matrices
• The Cauchy two-matrix model
• Universal microscopic correlation functions for products of independent Ginibre matrices
• Universal microscopic correlation functions for products of truncated unitary matrices
• Cauchy-Laguerre two-matrix model and the Meijer-G random point field
• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two--matrix model
• A note on the limiting mean distribution of singular values for products of two Wishart random matrices
• Gap probabilities for the Generalized Bessel process: a Riemann-Hilbert approach
• Matrices coupled in a chain. II. Spacing functions

Cited by in corpus (16)

• Recent exact and asymptotic results for products of independent random matrices
• Singular value statistics of matrix products with truncated unitary matrices
• Products of Independent Gaussian Random Matrices
• Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition
• Hard edge limit of the product of two strongly coupled random matrices
• Dropping the independence: singular values for products of two coupled random matrices
• The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $\theta =\frac{1}{2}$
• Bulk and soft-edge universality for singular values of products of Ginibre random matrices
• Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
• The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ is the reciprocal of an integer
• On the convergence of multi-level Hermite-Padé approximants
• Gaussian perturbations of hard edge random matrix ensembles
• Product matrix processes for coupled multi-matrix models and their hard edge scaling limits
• Large n limit for the product of two coupled random matrices
• Global rigidity and exponential moments for soft and hard edge point processes
• Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Padé Approximants and Asymptotics