
Contents
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Abstract Abstract
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15.1 Introduction: chain-matrix models 15.1 Introduction: chain-matrix models
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15.2 The Itzykson-Zuber Hermitian two-matrix model 15.2 The Itzykson-Zuber Hermitian two-matrix model
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15.2.1 Reduction to eigenvalues 15.2.1 Reduction to eigenvalues
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Determinantal point process and biorthogonal polynomials Determinantal point process and biorthogonal polynomials
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15.3 Biorthogonal polynomials: Christoffel–Darboux identities 15.3 Biorthogonal polynomials: Christoffel–Darboux identities
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15.3.1 Christoffel–Darboux identities 15.3.1 Christoffel–Darboux identities
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15.3.2 Differential equations 15.3.2 Differential equations
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Integral representation of the solutions: Riemann–Hilbert problem Integral representation of the solutions: Riemann–Hilbert problem
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Dual systems Dual systems
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15.4 The spectral curve 15.4 The spectral curve
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15.4.1 Mixed correlation functions and biorthogonal polynomials 15.4.1 Mixed correlation functions and biorthogonal polynomials
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The formula for mixed correlation functions The formula for mixed correlation functions
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15.5 Cauchy two-matrix models 15.5 Cauchy two-matrix models
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References References
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15 Two-matrix models and biorthogonal polynomials
Get accessMarco Bertola, Concordia University, Dept. of Math. & Stat., SGW Campus, LB-901-29, 1455 de Maisonneuve W., H3G 1M8, Montreal (QC), Canada, [email protected]
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Published:08 August 2018
Cite
Abstract
This article considers two cases of two-matrix models that are amenable to biorthogonal polynomials: Itzykson-Zuber interaction and Cauchy interaction. The features and applications of the biorthogonal polynomials relevant to either case are discussed, but first the article provides an overview of chain-matrix models. It then describes the Itzykson-Zuber Hermitian two-matrix model and the Christoffel–Darboux identities, along with the spectral curve. It also examines the so-called mixed correlation functions that are involved in the combinatorial applications of the two-matrix model before concluding with an analysis of the Cauchy two-matrix model, which in a ‘complication scale’ turns out to lie in between the one-matrix model and the Itzykson-Zuber two-matrix model.
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