Random matrices The GUE Wigner's semi-circle law Harer-Zagier Largest and smallest eigenvalues Lecture I: Asymptotics for large GUE random Summary: Random matrices The GUE Wigner's semi-circle law Harer-Zagier Largest and smallest eigenvalues Lecture I: Asymptotics for large GUE random matrices Steen Thorbjørnsen, University of Aarhus Random matrices The GUE Wigner's semi-circle law Harer-Zagier 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 × n-matrix 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 semi-circle law Harer-Zagier 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 Collections: Mathematics