Level Spacing Distribution Revisited - Theoretical Group Atomic ...

Gaussian Ensembles of Hermitian matrices H
Hermitian random matrix H = H †
consists of independent Gaussian entries
a) orthogonal ensemble, β = 1 - real random numbers,
b) unitary ensemble, β = 2 - complex numbers (real at the diagonal!)
c) symplectic ensemble, β = 4 - quaternions (2 × 2 matrices)
leading to 2N × 2N matrix with each eigenvalue occurring twice.
Different normalization conditions, we use the one implied by
the normal distribution 〈Hij〉 = 0 and σ2 = 〈H2 ij 〉 = 1
GUE ⇒ Unitary invariance, P(H) = P(UHU † )
leads to joint probability distribution (jpd) of eigenvalues xi
β P
−
Pβ(x1, . . .,xN) = CNe 2 j x2 j
|xj − xk| β
j