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On Airy solutions of P$_\mathrm{II}$ and the complex cubic ensemble of random matrices, II

Ahmad Barhoumi, Pavel Bleher, Alfredo Deaño, Maxim L. Yattselev

Abstract

We describe the pole-free regions of the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlevé equation, constructed from the Airy functions. This is achieved by exploiting the connection between these solutions and the recurrence coefficients of orthogonal polynomials that appear in the analysis of the ensemble of random matrices corresponding to the cubic potential.

On Airy solutions of P$_\mathrm{II}$ and the complex cubic ensemble of random matrices, II

Abstract

We describe the pole-free regions of the one-parameter family of special solutions of P, the second Painlevé equation, constructed from the Airy functions. This is achieved by exploiting the connection between these solutions and the recurrence coefficients of orthogonal polynomials that appear in the analysis of the ensemble of random matrices corresponding to the cubic potential.
Paper Structure (20 sections, 6 theorems, 100 equations, 16 figures)

This paper contains 20 sections, 6 theorems, 100 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Airy Solutions of Painlevé-II'
  3. Airy Solutions
  4. Hamiltonian Structure
  5. Tau Functions
  6. Cubic Random Matrix Model
  7. Connection to Airy Solutions
  8. Non-Hermitian Orthogonal Polynomials
  9. Phase Diagram
  10. Symmetric Contours
  11. Phase Diagram
  12. One-cut Cases
  13. Two-cut Cases
  14. Trefoil Cases
  15. Step 1
  16. ...and 5 more sections

Key Result

Theorem 3.2

Fix $N \geq 1$. Given $\lambda\in\overline\mathbb{C}$, it holds for each $n \in \mathbb{N}$ that where $q_n(z;\lambda)$ is a solution of P2 given by eq:q-1 and eq:backlund-forward, $p_n(z;\lambda)$ is the corresponding solution of P34, see ham_sys, while $\sigma_n(z;\lambda)$ is the corresponding solution of S2 determined by sigma1.

Figures (16)

  • Figure 1: Zeros (blue) and poles (red) of $q_3(z;0)$ (left panel) and $q_3(z;\infty)$ (right panel), see \ref{['eq:q-1']}--\ref{['eq:backlund-forward']} for the meaning of the second parameter.
  • Figure 2: Contours $L_0$, $L_1$ and $L_2$ used in the construction of $\Gamma$.
  • Figure 3: Schematic example of an admissible contour $T\in\mathcal{T}$ .
  • Figure 4: Schematic representation of (A) the critical graph $\mathcal{C}$; (B) the set $\Delta$ (solid lines) and the domain $\Omega_{(0)}$ (shaded region); (C) domain $O_{(0)}$ (shaded region).
  • Figure 5: Single cut domains $O_{0,0}$, $O_{0,\mathrm{i}}$, and $O_{0,-\mathrm{i}}$; double cut domains $O_{1,0}$, $O_{1,\mathrm{i}}$, $O_{1,-\mathrm{i}}$; and three cuts with a common endpoint domain $O_{1,-}$. Boundaries of these domains correspond to single cut cases.
  • ...and 11 more figures

Theorems & Definitions (18)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • ...and 8 more