arXiv 2201.128712022-01-30math-phmath.CAmath.MP

随机矩阵复立方系综中双切相区的研究

Investigation of the two-cut phase region in the complex cubic ensemble of random matrices

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

发表信息：: J. Math. Phys., 63, Paper No. 063303, 2022
DOI：: 10.1063/5.0086911

摘要：

研究了具有势$V(M)=-\frac{1}{3}M^3+tM$的随机矩阵的复立方幺正系综的相图，其中$t$是复参数。正如我们在前一篇论文中所证明的那样，模型的整个相空间$t\in\mathbb C$被划分为两个相区域，$O_{\mathsf{one-cut}}$和$O_{\mathsf{two-cut}}$，使得在$O_{\mathsf{one-cut}}$中平衡测度由一个Jordan弧（割）支撑，而在$O_{\mathsf{two-cut}}$中由两个割支撑。区域$O_{\mathsf{one-cut}}$和$O_{\mathsf{two-cut}}$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。在我们以前的工作中，单切相区进行了详细的研究。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的端点是解析函数的实部和虚部的参数$t$，但不是参数$t$本身。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。

英文摘要：

We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential $V(M)=-\frac{1}{3}M^3+tM$ where $t$ is a complex parameter. As proven in our previous paper, the whole phase space of the model, $t\in\mathbb C$, is partitioned into two phase regions, $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$, such that in $O_{\mathsf{one-cut}}$ the equilibrium measure is supported by one Jordan arc (cut) and in $O_{\mathsf{two-cut}}$ by two cuts. The regions $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$ are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In our previous work 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. 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.

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英文摘要：

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