We consider the problem of the exact computation of the marginal eigenvalue distributions in the Laguerre and Jacobi $\beta$ ensembles. In the case $\beta=1$ this is a question of long standing in the mathematical statistics literature. A recursive procedure to accomplish this task is given for $\beta$ a positive integer, and the parameter $\lambda_1$ a non-negative integer. This case is special due to a finite basis of elementary functions, with coefficients which are polynomials. In the Laguerre case with $\beta = 1$ and $\lambda_1 + 1/2$ a non-negative integer some evidence is given of their again being a finite basis, now consisting of elementary functions and the error function multiplied by elementary functions. Moreover, from this the corresponding distributions in the fixed trace case permit a finite basis of power functions, as also for $\lambda_1$ a non-negative integer. The fixed trace case in this setting is relevant to quantum information theory and quantum transport problem, allowing particularly the exact determination of Landauer conductance distributions in a previously intractable parameter regime. Our findings also aid in analyzing zeros of the generating function for specific gap probabilities, supporting the validity of an associated large $N$ local central limit theorem.

翻译：我们考虑Laguerre和Jacobi $β$ 系综中边缘特征值分布的精确计算问题。当 $β=1$ 时，这是数理统计文献中长期存在的问题。本文给出了一种递归程序来实现该目标，适用于 $β$ 为正整数且参数 $\lambda_1$ 为非负整数的情况。该情况的特殊性在于存在一个由初等函数构成的有限基，其系数为多项式。在 $β=1$ 且 $\lambda_1+1/2$ 为非负整数的Laguerre情形中，部分证据表明同样存在有限基，此时基函数由初等函数以及误差函数与初等函数的乘积组成。此外，由此可得固定迹情形下的相应分布允许由幂函数构成的有限基，这也适用于 $\lambda_1$ 为非负整数的情况。该设定下的固定迹情形与量子信息论和量子输运问题相关，能够精确确定先前难以处理的参数区域中的Landauer电导分布。我们的发现还有助于分析特定间隙概率生成函数的零点，支持相关大 $N$ 局部中心极限定理的有效性。