^{1}, M. Gekhtman

^{2}and J. Szmigielski

^{3}

### 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 polynomials.

This paper was completed at the Banff International Research Station. We thank BIRS for the hospitality and for providing excellent work conditions. We also thank Dima Korotkin for helpful discussions concerning spin bundles. M.B. and J.S. acknowledge a support by the Natural Sciences and Engineering Research Council of Canada. M.G. is partially supported by the National Science Foundation (NSF) Grant No. DMS-1101462.

I. INTRODUCTION

II. RIEMANN-HILBERT PROBLEM FOR CAUCHY BIORTHOGONAL POLYNOMIALS

III. THE FUNCTIONS

IV. DEIFT-ZHOU STEEPEST DESCENT ANALYSIS

A. Modifications of the Riemann-Hilbert problem

B. Outer parametrix

1. The abstract Riemann surface

2. Multiplier system χ and spinors

C. Theta-functional expressions

1. Genus 0 case

D. Local parametrix (solution of Problem 4.2)

1. The rank-two parametrix

E. Rank-three parametrix

F. Asymptotics of the biorthogonal polynomials

V. ASYMPTOTIC SPECTRAL STATISTICS AND UNIVERSALITY

### Key Topics

- Polynomials
- 25.0
- Jacobians
- 5.0
- Correlation functions
- 4.0
- Inequalities
- 3.0
- Lenses
- 3.0

## Figures

The supports of the equilibrium measures ρ_{1}, ρ_{2}.

The supports of the equilibrium measures ρ_{1}, ρ_{2}.

The jumps of the and functions in the gaps.

The jumps of the and functions in the gaps.

The final steps in the modification of the RHP.

The final steps in the modification of the RHP.

The modified jumps for Γ_{0} on a cut of the *right* side of the spectrum.

The modified jumps for Γ_{0} on a cut of the *right* side of the spectrum.

The modified jumps for Γ_{0} on a cut of the *left* side of the spectrum.

The modified jumps for Γ_{0} on a cut of the *left* side of the spectrum.

The RHP for the outer parametrix.

The RHP for the outer parametrix.

The contours of the residual RHP for the error term and the orders of the jumps matrices.

The contours of the residual RHP for the error term and the orders of the jumps matrices.

The Hurwitz diagram of the abstract spectral curve : the vertical dotted lines represent the identification of points.

The Hurwitz diagram of the abstract spectral curve : the vertical dotted lines represent the identification of points.

Our choice of the canonical homology basis on the abstract Riemann surface ; the solid lines represent arcs on Sheet 2, the dotted ones are arcs on Sheet 0, and the dashed ones are arcs on Sheet 1. In the example there are 5 total cuts and hence the genus is 3. We can declare that the curves lying on the same sheet are the α-cycles, whereas the curves lying on two different sheets are the corresponding β-cycles.

Our choice of the canonical homology basis on the abstract Riemann surface ; the solid lines represent arcs on Sheet 2, the dotted ones are arcs on Sheet 0, and the dashed ones are arcs on Sheet 1. In the example there are 5 total cuts and hence the genus is 3. We can declare that the curves lying on the same sheet are the α-cycles, whereas the curves lying on two different sheets are the corresponding β-cycles.

Depiction of the lifted contours . Their *X*-projection is just a closed path (in the picture we have chosen circles) intersecting the real axis only at *x* and one of *b* _{0} or *a* _{0}. The points *x*′, *y*′ belong to the gaps and by definition the curves lie on one sheet and are actually homologic to the α-cycles chosen before (see Fig. 9 and its caption).

Depiction of the lifted contours . Their *X*-projection is just a closed path (in the picture we have chosen circles) intersecting the real axis only at *x* and one of *b* _{0} or *a* _{0}. The points *x*′, *y*′ belong to the gaps and by definition the curves lie on one sheet and are actually homologic to the α-cycles chosen before (see Fig. 9 and its caption).

Depiction of the three sheets and how they are mapped onto three disjoint regions of the *t*-plane. The real *x*-axis is mapped to the real *t*-axis and the two sides of each cut are mapped to the boundaries of the oval-shaped regions. The intersection of the ovals with the real *t*-axis are the four ramification points of the map *z*(*t*).

Depiction of the three sheets and how they are mapped onto three disjoint regions of the *t*-plane. The real *x*-axis is mapped to the real *t*-axis and the two sides of each cut are mapped to the boundaries of the oval-shaped regions. The intersection of the ovals with the real *t*-axis are the four ramification points of the map *z*(*t*).

The exact Riemann-Hilbert problem near a right endpoint in , and in terms of the zooming local coordinate ξ.

The exact Riemann-Hilbert problem near a right endpoint in , and in terms of the zooming local coordinate ξ.

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