Investigation of the two-cut phase region in the complex cubic ensemble of random matrices
Investigation of the two-cut phase region in the complex cubic ensemble of random matrices
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DOI：
10.1063/5.0086911
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发表时间：
2022-01
影响因子：
1.3
通讯作者：
A. Barhoumi;P. Bleher;A. Deaño;M. Yattselev
中科院分区：
文献类型：
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作者：
A. Barhoumi;P. Bleher;A. Deaño;M. Yattselev
我们研究了具有势的随机矩阵的复立方幺正系综的相图[公式：见正文]，其中t是一个复参数。正如我们以前的论文[Bleher等人，J. Stat. 166，784-827（2017）]，模型的整个相空间[公式：见正文]被划分为两个相区域，[公式：见正文]和[公式：见正文]，使得在[公式：见正文]中，平衡测度由一个乔丹弧（切割）支持，而在[公式：见正文]中由两个切割支持。区域[公式：见正文]和[公式：[见正文]由临界曲线分开，临界曲线可以根据辅助二次微分的临界轨迹来计算。在Bleher等人[J. Stat. 166，784-827（2017）]，详细研究了单切相区域。在本文中，我们研究的两个切割区域。我们证明了在两个切割区域中，切割的端点是参数t的真实的和虚部的解析函数，但不是参数t本身的解析函数（因此，柯西-黎曼方程的端点是违反的）。我们还得到了与随机矩阵系综相关的正交多项式的半经典渐近性及其递推系数。证明是基于Riemann-Hilbert方法的半经典渐近的正交多项式和理论的S-曲线和二次微分。
We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential [Formula: see text], where t is a complex parameter. As proven in our previous paper [Bleher et al., J. Stat. Phys. 166, 784–827 (2017)], the whole phase space of the model, [Formula: see text], is partitioned into two phase regions, [Formula: see text] and [Formula: see text], such that in [Formula: see text] the equilibrium measure is supported by one Jordan arc (cut) and in [Formula: see text] by two cuts. The regions [Formula: see text] and [Formula: see text] are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In Bleher et al. [J. Stat. Phys. 166, 784–827 (2017)], the one-cut phase region was investigated in detail. In the present paper, we investigate the two-cut region. We prove that in the two-cut region, the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter t, but not of the parameter t itself (so that the Cauchy–Riemann equations are violated for the endpoints). We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann–Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of S-curves and quadratic differentials.