The Cauchy two-matrix model

Abstract

We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann-Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. A steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitean matrix model is related to a hyperelliptic curve.

34 pages, 2 figures. V2: changes only to metadata

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Cited by in corpus (35)

• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits
• Cauchy-Laguerre two-matrix model and the Meijer-G random point field
• Relating the Bures measure to the Cauchy two-matrix model
• Products of Independent Gaussian Random Matrices
• Universality conjecture and results for a model of several coupled positive-definite matrices
• Bures-Hall Ensemble: Spectral Densities and Average Entropies
• Probability of all eigenvalues real for products of standard Gaussian matrices
• The Cauchy two-matrix model, C-Toda lattice and CKP hierarchy
• Degasperis-Procesi peakon dynamical system and finite Toda lattice of CKP type
• Muttalib--Borodin ensembles in random matrix theory --- realisations and correlation functions
• Hard edge limit of the product of two strongly coupled random matrices
• A view of the peakon world through the lens of approximation theory
• Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two--matrix model
• An Application of Pfaffians to multipeakons of the Novikov equation and the finite Toda lattice of BKP type
• Entropy fluctuation formulas of fermionic Gaussian states
• Peakons and Cauchy Biorthogonal Polynomials
• Cubic String Boundary Value Problems and Cauchy Biorthogonal Polynomials
• Average characteristic polynomials in the two-matrix model
• Quantum interpolating ensemble: Biorthogonal polynomials and average entropies
• Permanental processes from products of complex and quaternionic induced Ginibre ensembles
• Inverse problems associated with integrable equations of Camassa-Holm type; explicit formulas on the real axis, I
• Product matrix processes for coupled multi-matrix models and their hard edge scaling limits
• New recursive residue formulas for the topological expansion of the Cauchy Matrix Model
• Rank shift conditions and reductions of 2d-Toda theory
• Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
• Regularity of a vector potential problem and its spectral curve
• Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles
• The Hermitian two matrix model with an even quartic potential
• Discrete integrable systems and condensation algorithms for Pfaffians
• Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices
• Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis
• Two-parameter generalisations of Cauchy bi-orthogonal polynomials and integrable lattices
• Cauchy Biorthogonal Polynomials
• Fox H-kernel and $\theta$-deformation of the Cauchy two-matrix model and Bures ensemble