We study the probability that all the eigenvalues of n × n Hermitian matrices, from the Laguerre unitary ensemble with the weight $x^{\gamma }{\mathrm{e}}^{-4nx},x\in \left[0,\infty \right),\gamma >-1$, lie in the interval [0, α]. By using previous results for finite n obtained by the ladder operator approach of orthogonal polynomials, we derive the large n asymptotics of the largest eigenvalue distribution function with α ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159, 151–174 (1994)] and later proved by Deift, Its, and Krasovsky [Commun. Math. Phys. 278, 643–678 (2008)]. Our conclusions are reduced to those of Deift et al. when γ = 0. It should be pointed out that our derivation is straightforward but not rigorous, and hence, the above results are stated as conjectures.

中文翻译：

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Laguerre幺正系综最大特征值分布的渐近线

我们研究了n × n Hermitian 矩阵的所有特征值的概率，来自具有权重的 Laguerre 酉系综$X^{\gamma }{\mathrm{电子}}^{-4nX},X\in \left[0,\infty \right),\gamma >-1$, 位于区间 [0, α ] 内。通过使用正交多项式梯形算子方法获得的有限n 的先前结果，我们推导出最大特征值分布函数的大n渐近线，其中α范围从 0 到软边缘。此外，在软边缘，我们计算了 Tracy 和 Widom [Commun. 数学。物理。159 , 151–174 (1994)]，后来被 Deift、Its 和 Krasovsky 证明 [Commun. 数学。物理。278 , 643–678 (2008)]。我们的结论被简化为 Deift等人的结论。当γ = 0. 需要指出的是，我们的推导是直接的，但并不严谨，因此，上述结果为猜想。