Book Review: Combinatorics and random matrix theory; Dynamical approach to random matrix theory
Terence Tao · Bulletin of the American Mathematical Society · 2019
A fundamental phenomenon in the statistical study of large complex systems is that of universality: in the limit where the number N of components of the system goes to infinity, the distribution of various natural statistics of that system, after suitable normalisation, will often converge (in a suitable probabilistic sense) to a universal limiting distribution, the nature of which is largely independent of the microscopic features of the individual components of such a system.For instance, the two most fundamental theorems in probability can both be interpreted as basic examples of universality:• Law of large numbers.If X 1 , X 2 , . . .are independent and identically distributed (or iid for short) real random variables of finite first moment E|X i | < ∞ with the normalisation EX i = 0, then the normalised aver-converge in probability to the deterministic constant 0.• Central limit theorem.If X 1 , X 2 , . . .are iid real random variables of finite second moment E|X i | 2 < ∞ with the normalisation EX i = 0, EX 2 i = 1, then the normalised averages