经典矩阵系综的某些参数相关推广的解释
Interpretations of some parameter dependent generalizations of classical matrix ensembles
摘要：
利用概率密度函数定义了经典矩阵系综的两类参数依赖的推广。当参数变化时，一个在具有正交对称性的两个经典系综的叠加的本征值PDF和具有酉对称性的单个经典系综的本征值PDF之间插值，而另一个在具有正交对称性的经典系综和具有辛对称性的经典系综之间插值。我们给这些PDF的解释相关的概率矩阵之间的连续罗宾逊-Schensted-Knuth对应关系，从某些指数分布中选择的条目，和非相交的晶格路径，并在这个过程中的概率措施分区和对分区确定。后者一般通过使用麦克唐纳多项式理论，和一个特定的连续极限-Jacobi极限-所产生的措施，给出PDF相关的安德森的工作中出现的Selberg积分。通过将安德森的工作解释为给出某个有理函数的零点的PDF，然后可以识别其特征值PDF实现原始参数相关PDF的随机矩阵。这条理论允许采样的原始参数依赖PDF，他们的安德森型推广和相关的边际分布，从零的随机三项递归定义的某些多项式。
英文摘要：
Two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions (PDFs). As the parameter is varied, one interpolates between the eigenvalue PDF for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue PDF for a single classical ensemble with unitary symmetry, while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry. We give interpretations of these PDFs in terms of probabilities associated to the continuous Robinson-Schensted-Knuth correspondence between matrices, with entries chosen from certain exponential distributions, and non-intersecting lattice paths, and in the course of this probability measures on partitions and pairs of partitions are identified. The latter are generalized by using Macdonald polynomial theory, and a particular continuum limit -- the Jacobi limit -- of the resulting measures is shown to give PDFs related to those appearing in the work of Anderson on the Selberg integral. By interpreting Anderson's work as giving the PDF for the zeros of a certain rational function, it is then possible to identify random matrices whose eigenvalue PDFs realize the original parameter dependent PDFs. This line of theory allows sampling of the original parameter dependent PDFs, their Anderson-type generalizations and associated marginal distributions, from the zeros of certain polynomials defined in terms of random three term recurrences.
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