Lectures on Random Matrix Theory for Course at SAMSI
Peter D. Miller · 2008
where ZN is a normalization constant (partition function) and dM denotes Lebesgue measure on the algebraically independent real components of M: j and Mii ∈ R. Some notes: 1. V (x) is a potential increasing sufficiently rapidly for large |x| to make the measure normalizable. V (M) is defined through the spectral theorem: for each Hermitian matrix M there exists a unitary matrix U such that M = Udiag(x1, . . . , xN )U†, where x1 ≤ x2 ≤ · · · ≤ xN are the (real) eigenvalues of M. Then V (M) is the matrix V (M) := Udiag(V (x1), . . . , V (xN ))U† . (2) As the trace is invariant under conjugation, note that