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 \({p \geq 2}\) 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.

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Correspondence to Thomas Bothner.

Additional information

The first author is supported in part by the Natural Sciences and Engineering Research Council of Canada.

The second author acknowledges support by Concordia University through a postdoctoral fellow top-up award.

Communicated by P. Deift

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Bertola, M., Bothner, T. Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices. Commun. Math. Phys. 337, 1077–1141 (2015). https://doi.org/10.1007/s00220-015-2327-7

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Keywords

  • Spectral Curve
  • Meijer
  • Equilibrium Measure
  • Hilbert Problem
  • Correlation Kernel