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Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model
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View: Figures


Image of FIG. 1.
FIG. 1.

The supports of the equilibrium measures ρ1, ρ2.

Image of FIG. 2.
FIG. 2.

The jumps of the and functions in the gaps.

Image of FIG. 3.
FIG. 3.

The final steps in the modification of the RHP.

Image of FIG. 4.
FIG. 4.

The modified jumps for Γ0 on a cut of the right side of the spectrum.

Image of FIG. 5.
FIG. 5.

The modified jumps for Γ0 on a cut of the left side of the spectrum.

Image of FIG. 6.
FIG. 6.

The RHP for the outer parametrix.

Image of FIG. 7.
FIG. 7.

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

Image of FIG. 8.
FIG. 8.

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

Image of FIG. 9.
FIG. 9.

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.

Image of FIG. 10.
FIG. 10.

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).

Image of FIG. 11.
FIG. 11.

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).

Image of FIG. 12.
FIG. 12.

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


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model