论文标题
Bures-hall合奏和Cauchy-Laguerre两矩阵模型的间隙概率
Gap probabilities for the Bures-Hall Ensemble and the Cauchy-Laguerre Two-Matrix Model
论文作者
论文摘要
可以说，可以说研究量子机械密度矩阵的光谱，而没有对系统的知识，可以说是研究量指标和相关的井孔测量。我们研究了该模型频谱中差距的可能性，无论是在底部$ [0，s）$还是在top $（s，1] $处，利用了这种pfaffian point-pococess与同盟问题的连接在二维cauchy-laguerre bi-ordornogonal polynomial系统的确定性点过程中，现在$ ye $ weardor $与拉格尔密度相比，关于凯奇双交的多项式系统的新一般结果：尤其是一个新的基督教式 - 佛教级公式，重现核心和差异方程式，用于多项式及其相关的功能。重量的拉瓜类型密度，与考奇 - 拉瓜系统的直接相关性，我们在两个变形变量$ s，t $中构建了一个封闭的约束，非线性微分方程的封闭系统，并观察到复发，光谱和变形衍生结构形成了与Laxe equations的兼容和可集成的三分线。
The Bures metric and the associated Bures-Hall measure is arguably the best choice for studying the spectrum of the quantum mechanical density matrix with no apriori knowledge of the system. We investigate the probability of a gap in the spectrum of this model, either at the bottom $ [0,s) $ or at the top $ (s,1] $, utilising the connection of this Pfaffian point-process with the allied problem in the determinantal point-process of the two-dimensional Cauchy-Laguerre bi-orthogonal polynomial system, now deformed with two variables $s,t$. To this end we develop new general results about Cauchy bi-orthogonal polynomial system for a more general class of weights than the Laguerre densities: in particular a new Christoffel-Darboux formula, reproducing kernels and differential equations for the polynomials and their associated functions. This system is most simply expressed as rank-3 matrix variables and possesses an associated cubic bilinear form. Furthermore under specialisation to truncated Laguerre type densities for the weight, of direct relevance to the Cauchy-Laguerre system, we construct a closed system of constrained, nonlinear differential equations in two deformation variables $s,t$, and observe that the recurrence, spectral and deformation derivative structures form a compatible and integrable triplet of Lax equations.
