Universality in Random Matrix Theory for orthogonal and symplectic ensembles
Universality in Random Matrix Theory for orthogonal and symplectic ensembles
复制标题
正交系综和辛系综随机矩阵理论的普适性
DOI：
10.1093/imrp/rpm004
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
Dimitri Gioev
中科院分区：
文献类型：
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作者：
P. Deift;Dimitri Gioev
本文证明了随机矩阵正交系综和辛系综在一类权函数w（x）=exp（-V（x））的标度极限下的普适性猜想，其中V（x）=kappa_{2 m}x^{2 m}+. kappa_{2m}>0.对于这样的权重，在单个区间上支持相关联的均衡度量。我们的结果的精确陈述在下面的定理1.1中给出。关于酉系综的普适性猜想的证明，对于同一类权重，参见[DKMVZ 2]。
我们的出发点是正交和辛相关核的Widom表示[W]，根据酉情况下产生的核加上校正项，该校正项由正交多项式（OP）{p_j（x）}，j= 0，1，.，关于权重w（x）。[W]中的计算反过来又依赖于Tracy和Widom [TW 2]的早期工作。结果是（见[W]和下面的定理2.1和2.2），只有在范围j=N+O（1），N->无穷大的OP对校正项有贡献。在控制这个校正项，并因此证明普遍性的正交和辛的情况下，统一Plancherel-Rotach型渐近的OP的[DKMVZ 2]中发现发挥了重要作用，但有显着的新的分析困难，必须克服这是不存在的单一的情况下。我们注意到，我们不使用斜正交多项式。
We give a proof of the Universality Conjecture for orthogonal and symplectic ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial, V(x)=kappa_{2m}x^{2m}+..., kappa_{2m}>0. For such weights the associated equilibrium measure is supported on a single interval. The precise statement of our results is given in Theorem 1.1 below. For a proof of the Universality Conjecture for unitary ensembles, for the same class of weights, see [DKMVZ2].
Our starting point is Widom's representation [W] of the orthogonal and symplectic correlation kernels in terms of the kernel arising in the unitary case plus a correction term which is constructed out of derivatives and integrals of orthonormal polynomials (OP's) {p_j(x)}, j=0,1,..., with respect to the weight w(x). The calculations in [W] in turn depend on the earlier work of Tracy and Widom [TW2]. It turns out (see [W] and also Theorems 2.1 and 2.2 below) that only the OP's in the range j=N+O(1), N->infinity, contribute to the correction term. In controlling this correction term, and hence proving Universality for both the orthogonal and symplectic cases, the uniform Plancherel--Rotach type asymptotics for the OP's found in [DKMVZ2] play an important role, but there are significant new analytical difficulties that must be overcome which are not present in the unitary case. We note that we do not use skew orthogonal polynomials.