Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices
Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices
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DOI：
10.1007/s00220-015-2327-7
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发表时间：
2014-07
影响因子：
2.4
通讯作者：
M. Bertola;Thomas Bothner
中科院分区：
文献类型：
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作者：
M. Bertola;Thomas Bothner
本文包含两个主要部分：第一部分，我们分析了柯西相互作用链耦合矩阵的一般情况。与 Itzykson-Zuber 相互作用模型类似，柯西链的特征值形成多级行列式点过程。我们首先根据柯西双正交多项式计算所有相关函数，并将它们定位为黎曼-希尔伯特问题的矩阵值解的特定条目。在第二部分中，我们将外部势固定为经典拉盖尔权重。然后，当平衡测度的支持包含原点时，我们导出柯西双正交多项式的强渐近性。结果，我们获得了多级随机行列式点场的一个新的普适类族，其中包括 1 级的 Bessel 普适性和 2 级的 Meijer-G 普适性。我们的分析使用 Deift-Zhou 非线性最速下降法和根据 Meijer G 函数显式构造原点参数矩阵。完整的黎曼-希尔伯特问题的解仅在 p=3 时严格推导，但证明的一般框架可以扩展到任意长度 p 的柯西链。
The paper contains two main parts: in the first part, we analyze the general case ofmatrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of amatrix valued solution of a Riemann–Hilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multi-level random determinantal point fields, which include the Besseluniversality for 1-level and the Meijer-G universality for 2-level. Our analysis uses the Deift-Zhou nonlinear steepest descent method and the explicit construction of aorigin parametrix in terms of Meijer G-functions. The solution of the full Riemann–Hilbert problem is derived rigorously only forp= 3 but the general framework of the proof can be extended to the Cauchy chain of arbitrary lengthp.