Large deviations of spread measures for Gaussian matrices
Large deviations of spread measures for Gaussian matrices
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DOI：
10.1088/1742-5468/2016/04/043306
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发表时间：
2014-03
期刊：
影响因子：
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通讯作者：
F. D. Cunden;P. Vivo
中科院分区：
文献类型：
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作者：
F. D. Cunden;P. Vivo
对于大型 n×m 高斯矩阵，我们计算广义方差和总方差的联合统计量，包括大偏差尾部 - 相应 n×n 协方差矩阵的缩放对数行列式 H 和迹 T。使用库仑气体技术，我们发现它们的联合分布 Pn(h,t) 的拉普拉斯变换对于大 n、m（c=m/n⩾1 固定）衰减为 P^n(s,w)≈exp(−βn2J(s,w))，其中 β 是系综的戴森指数，J(s, w) 是与 β 无关的大偏差函数，我们可以针对任何 c 精确计算它。通过大量的数值模拟计算并检查了实际空间中相应的大偏差函数。结果通过基于 Laguerre-Selberg 积分的有限 n, m 处理得到补充。非典型小对数行列式的统计显示是由最小特征值的分裂驱动的，导致大偏差速度的突然变化。
For a large n×m Gaussian matrix, we compute the joint statistics, including large deviation tails, of generalized and total variance—the scaled log-determinant H and trace T of the corresponding n×n covariance matrix. Using a Coulomb gas technique, we find that the Laplace transform of their joint distribution Pn(h,t) decays for large n, m (with c=m/n⩾1 fixed) as P^n(s,w)≈exp(−βn2J(s,w)), where β is the Dyson index of the ensemble and J(s, w) is a β-independent large deviation function, which we compute exactly for any c. The corresponding large deviation functions in real space are worked out and checked with extensive numerical simulations. The results are complemented with a finite n, m treatment based on the Laguerre–Selberg integral. The statistics of atypically small log-determinants is shown to be driven by the split-off of the smallest eigenvalue, leading to an abrupt change in the large deviation speed.