Department of Quantum Physics and Photonics, Institute of Physics, UNAM, P.O. Box 20-364, 01000 Mexico City, Mexico and London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, United Kingdom.
Institute of Physics, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil; Physics Department, Federal University of Santa Maria, 97105-900 Santa Maria, Brazil; and London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, United Kingdom.
Phys Rev E. 2018 Mar;97(3-1):032124. doi: 10.1103/PhysRevE.97.032124.
Wishart random matrices with a sparse or diluted structure are ubiquitous in the processing of large datasets, with applications in physics, biology, and economy. In this work, we develop a theory for the eigenvalue fluctuations of diluted Wishart random matrices based on the replica approach of disordered systems. We derive an analytical expression for the cumulant generating function of the number of eigenvalues I_{N}(x) smaller than x∈R^{+}, from which all cumulants of I_{N}(x) and the rate function Ψ_{x}(k) controlling its large-deviation probability Prob[I_{N}(x)=kN]≍e^{-NΨ_{x}(k)} follow. Explicit results for the mean value and the variance of I_{N}(x), its rate function, and its third cumulant are discussed and thoroughly compared to numerical diagonalization, showing very good agreement. The present work establishes the theoretical framework put forward in a recent letter [Phys. Rev. Lett. 117, 104101 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.104101] as an exact and compelling approach to deal with eigenvalue fluctuations of sparse random matrices.
具有稀疏或稀释结构的 Wishart 随机矩阵在处理大型数据集方面无处不在，其应用领域涵盖物理、生物和经济。在这项工作中，我们基于无序系统的复形方法，为稀释 Wishart 随机矩阵的特征值波动发展了一种理论。我们推导出了特征值个数 I_{N}(x)小于 x∈R^{+}的累积生成函数的解析表达式，从中可以得到 I_{N}(x)的所有累积量和控制其大偏差概率 Prob[I_{N}(x)=kN]≍e^{-NΨ_{x}(k)}的速率函数 Ψ_{x}(k)。我们讨论并彻底比较了 I_{N}(x)的平均值和方差、其速率函数及其三阶累积量的显式结果，并与数值对角化进行了比较，结果非常吻合。这项工作为最近的一封信[Phys. Rev. Lett. 117, 104101 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.104101]中提出的理论框架建立了理论基础，该框架是处理稀疏随机矩阵特征值波动的一种精确而有力的方法。
