An Introduction to Random Matrices

Abstract

The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

On the spectral norm of Gaussian random matrices

“…On the other hand, applying Theorem 3.2 with d 1 = d 2 = 4 immediately yields the analogous bound for X (1) . We therefore finally obtain…”

The Tracy–Widom distribution is not infinitely divisible

“…Furthermore it can be deduced from [31], [32], [34] and [4] (see also [2,Exercise 3.9.36]) that the asymptotics for F β (x) as x → ∞, for β = 1, 2 or 4 is,…”

“…where k N,β is a non negative constant and for β = 1, 2 or 4, it can be computed by Selberg's Integral formula (see [2,Theorem 2.5.8]). The PDF (1) exhibits strong dependence of the eigenvalues of the G(O/U/S)E ensembles.…”

“…Investigate the infinite divisibility of λ max H β N in the tridiagonal β-Hermite ensemble (2). The article [16] it may be useful.…”

Random Matrix Theory and Its Innovative Applications

“…It is known that if we take a matrix A=randn(m,n) (in MATLAB notation), which is an m × n random matrix with . Thus if we have a correlation matrix with eigenvalues larger than (1 + √ r) 2 we consider these as signal rather than noise. Theoretical understanding of the "signal" eigenvalues may be found in [19].…”