arXiv 1004.34792010-04math.PRmath.FAmath.OA

高斯酉包络的渐近展开

Asymptotic expansions for the Gaussian Unitary Ensemble

Uffe Haagerup, Steen Thorbjørnsen

备注：: References have been added

摘要：

设g：{ \mathbb R} --> { \mathbb C}是所有导数有界的C^{ \infty }-函数，tr_n表示n × n矩阵上的正规化迹. Ercolani和McLaughlin在文献[EM]中对一类相当一般的随机矩阵X_n，包括高斯酉矩阵（GUE），建立了均值E{tr_n（g（X_n））}的渐近展开。使用分析的方法，我们在本文中提供了一个替代的证明，这种渐近展开的GUE的情况下。具体地，我们推导出对于GUE随机矩阵X_n，E{tr_n（g（X_n））}= \frac{1}{2π} \int_{-2}^2 g（x）\sqrt{4-x^2} dx + \sum_{j=1}^k \frac{α_j(g)}{n^{2j}} + O（n^{-2k-2}），其中k是任意正整数。考虑g的映射，我们确定系数α_j（g），j \in{\mathbb N}为L. Schwarts）。我们得到了协方差Cov{Tr_n[f（X_n）]，Tr_n[g（X_n）]}的一个类似的渐近展开式，其中f是与g同类的函数，Tr_n= ntr_n.特别关注的情况下，g（x）=1/（z-x）和f（x）=1/（w-x）的非实复数z和w。在这种情况下，上面考虑的均值和协方差分别对应于GUE（n，1/n）的一维和二维柯西（或斯蒂尔吉斯）变换。

英文摘要：

Let g:{\mathbb R} --> {\mathbb C} be a C^{\infty}-function with all derivatives bounded and let tr_n denote the normalized trace on the n x n matrices. In the paper [EM] Ercolani and McLaughlin established asymptotic expansions of the mean value E{tr_n(g(X_n))} for a rather general class of random matrices X_n,including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a GUE random matrix X_n that E{tr_n(g(X_n))}= \frac{1}{2π}\int_{-2}^2 g(x)\sqrt{4-x^2} dx +\sum_{j=1}^k\frac{α_j(g)}{n^{2j}}+ O(n^{-2k-2}), where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients α_j(g), j\in{\mathbb N}, as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Tr_n[f(X_n)],Tr_n[g(X_n)]}, where f is a function of the same kind as g, and Tr_n=n tr_n. Special focus is drawn to the case where g(x)=1/(z-x) and f(x)=1/(w-x) for non-real complex numbers z and w. In this case the mean and covariance considered above correspond to, respectively, the one- and two-dimensional Cauchy (or Stieltjes) transform of the GUE(n,1/n).

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