特征值的最近间距
Closest Spacing of Eigenvalues
摘要：
研究了n × n随机酉矩阵的最小特征值间距的分布。最小间距尺度为$n^{-4/3}$，而不是$n^{-2}$，如单位圆上的n个独立点的情况，说明了随机矩阵的特征值“相互排斥”的众所周知的现象。我们推导出在极限中重新标度的最小间距的分布为$n\to\infty$。为了找到最小间距，我们计算比$xn^{-4/3}$更接近的特征值对的数量。我们使用概率论来猜测这个整数值随机变量是泊松，计算极限分布的实际矩，并发现实际矩与猜测的矩相匹配。匹配矩证明了启发式猜测的正确性，并直接导出了主要结果。我们证明了类似的结果高斯酉系综（GUE），并与限制，一类普遍的酉系综（UUE）研究Deift，Kreicherbauer，McLaughlin，Venakides和周。
英文摘要：
We study the distribution of the minimum spacing between eigenvalues of a random n by n unitary matrix. The minimum spacing scales as $n^{-4/3}$, not $n^{-2}$ as would be the case for n independent points on the unit circle, illustrating the well known phenomenon that the eigenvalues of random matrices 'repel each other'. We derive the distribution for the rescaled minimum spacing in the limit as $n\to\infty$. To find the minimum spacing, we count the number of eigenvalue pairs closer than $xn^{-4/3}$. We use heuristics to guess that this integer-valued random variable is Poisson, calculate the actual moments of the limiting distribution, and find that the actual moments match those of the guess. The matching moments prove that the heuristic guess is correct, and lead directly to the main result. We prove analogous results for the Gaussian unitary ensemble (GUE) and, with restrictions, a universal class of unitary ensembles (UUE) studied by Deift, Kreicherbauer, McLaughlin, Venakides, and Zhou.
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