 
Summary: Random matrices The GUE Wigner's semicircle law HarerZagier Largest and smallest eigenvalues
Lecture I: Asymptotics for large GUE random
matrices
Steen Thorbjørnsen, University of Aarhus
Random matrices The GUE Wigner's semicircle law HarerZagier Largest and smallest eigenvalues
Random Matrices
Definition. Let (, F, P) be a probability space and let n be a
positive integer. Then a random n × n matrix A on (, F, P) is an
n × nmatrix
A = (aij )1i,jn,
where all the entries are complex valued random variables on
(, F, P). In other words, A is a measurable mapping
A: (, F, P) (Mn(C), B(Mn(C))),
when Mn(C) is equipped with its Borel algebra B(Mn(C)).
Random matrices The GUE Wigner's semicircle law HarerZagier Largest and smallest eigenvalues
The spectral distribution of a selfadjoint random
matrix
Let A: Mn(C) be a selfadjoint random matrix, i.e.,
A() = A() for all . Then for each we consider the ordered
eigenvalues
