Random Matrices and the Statistical Theory of Energy Levels

By M. L. Mehta, E. Merzbacher. 2014

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Random matrices: tail bounds for gaps between eigenvalues (2015, methodology): “For the model Xn , by Wigner’s semi-circle law [21], most eigenvalues are in the interval [−2√n, 2√n], thus the average gap is of order n−1/2.”
Random matrices: Universality of local spectral statistics of non-Hermitian matrices (2012, background): “one (or in other words, the probability distribution of each ˘ ij is 1 ˇ e jz2 dz, and the real and imaginary parts of ˘ ij independently have the distribution N(0;1=2) R). As is well known (see e.g. [35]), the correlation functions of a complex gaussian matrix are given by the explicit formula (2) ”
Universality and the circular law for sparse random matrices. (2010, background): “The nonsparse circular law has been proven in many special cases by many authors, including Mehta [20] (Gaussian case), Girko [13, 14], Edelman [11] (real Gaussian case), Bai [2] and Bai and Silverstein [1] [continuous case with bounded (2+ δ)th moment, for δ > 0], Götze and Tikhomirov [15] (sub-Gau”
The Elliptic Law (2012, unknown): “[28] M. L. Mehta, Random Matrices, third edition.”