Abstract: In the random matrix theory, the 1-level correlation functions R¹_nβ(x) ( β is generally called Dyson's index) can be explained as the distribution density of energy levels which can be found nearby x. As yet, its limit behavior is still remarkably noticed by many mathematicians and physicists. In the case of unitary ensemble (i.e. β=2), the 1-level correlation functions R¹_n2(x) is closely related to the weight function μ(x) in the classical orthogonal polynomial theory. For the simplication of statement, only the Gaussian unitary ensemble case is considered. The authors find that in weak sense, the limit behavior of R¹_n2(x) is not disturbed by appending a "good" nontrivial multiplicative factor to μ(x).

Key words: Gaussian unitary ensembles, correlation functions, moments, laguerre polynomials

摘要： 在随机矩阵理论中，1-级相关函数 R¹_nβ(x)（β一般称为Dyson指标）可以用来刻画在 x 附近能够发现能级的分布密度。直到现在，它的极限行为仍然受到许多数学家和物理学家的关注。在酉系综情形下（即 β=2），1-级相关函数 R¹_n2(x) 与经典多项式理论中的权函数 μ(x) 密切相关。文中，为叙述上简单起见，只考虑Gauss酉系综的情形。可以发现，在弱收敛意义下，给权函数 μ(x) 以“好的”而非平凡的乘积因子，1-级相关函数 R¹_n2(x) 的极限行为不受干扰。

关键词: Gauss酉系综, 相关函数, 矩, Laguerre多项式