2013
DOI: 10.1063/1.4802455
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Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model

Abstract: We apply the nonlinear steepest descent method to a class of 3 × 3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model. The general case of two equilibrium measures supported on an arbitrary number of intervals is considered. In this case, we solve the Riemann-Hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three-sheeted Riemann surface of arbitrary genus and establish strong asymptotic results for the Cauchy biorthogonal polyn… Show more

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Cited by 17 publications
(26 citation statements)
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“…which follows from the expression in Theorems 2.1, 5.5, 5.6 (see (5)(6), (5)(6)(7)(8)(9)). For large n (in fact even for small n's) and x = ζ n 2 n n+1…”
Section: Applicationsmentioning
confidence: 90%
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“…which follows from the expression in Theorems 2.1, 5.5, 5.6 (see (5)(6), (5)(6)(7)(8)(9)). For large n (in fact even for small n's) and x = ζ n 2 n n+1…”
Section: Applicationsmentioning
confidence: 90%
“…It is thus an important step to present the kernels in a form similar to (3)(4)(5)(6)(7)(8)(9). This is the purpose of the expressions in Proposition 5.2.…”
Section: Outlook: Computation Of the Gap Probabilities And Integrablementioning
confidence: 99%
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“…Another class of examples consists of multimatrix models of positive Hermitian matrices subject to the Cauchy interaction. This class of multi-matrix models was introduced in Bertola, Gekhtman, and Szmigielski [12], and studied further in Bertola, Gekhtman, and Szmigielski [13,14], and in Bertola and Bothner [10]. For other examples of multi-matrix models, and for an explanation of their relevance to quantum field theory and to statistical mechanics we refer the reader to Eynard, Kimura, and Ribault [17, Section 2.2.…”
Section: Introductionmentioning
confidence: 99%
“…In particular, in [21] it was shown that for the Novikov's equation, Cauchy biorthogonal polynomials [6] are the solutions to the approximation problem relevant to the inverse problem. Recent developments (see [5], [4], [7], [8] and [3]) suggest that Cauchy biorthogonal polynomials can be also useful in the investigation of various problems in random matrix theory. In this paper it is shown that Cauchy biorthogonal polynomials [6] can be used to solve the peakon inverse problem arising in the DP equation.…”
Section: Introductionmentioning
confidence: 99%