Gaussian Orthogonal Ensemble

introduction

• The Gaussian orthogonal ensemble GOE(n) is described by the Gaussian measure with density

\[ \frac{1}{Z_{\text{GOE}(n)}} e^{- \frac{n}{4} \mathrm{tr} H^2} \] on the space of \(n\times n\) real symmetric matrices \(H=(H_{ij})\)

• Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry

level spacing of eigenvalues

• From the ordered sequence of eigenvalues \(\lambda_1 < \ldots < \lambda_n < \lambda_{n+1} < \ldots\), one defines the normalized spacings \(s = (\lambda_{n+1} - \lambda_n)/\langle s \rangle\), where \(\langle s \rangle =\langle \lambda_{n+1} - \lambda_n \rangle\) is the mean spacing.
• The probability distribution of spacings is approximately given by,

\[ p_1(s) = \frac{\pi}{2}s\, \mathrm{e}^{-\frac{\pi}{4} s^2} \] for the orthogonal ensemble GOE \(\beta=1\)

joint eigenvalue distribution

• The joint probability density for the eigenvalues λ₁,λ₂,...,λ_n of GOE is given by

\[\frac{1}{Z_{\beta, n}} \prod_{k=1}^n e^{-\frac{\beta n}{4}\lambda_k^2}\prod_{i<j}\left|\lambda_j-\lambda_i\right|^\beta~, \quad (1)\] where the Dyson index \(\beta=1\), counts the number of real components per matrix element; Z_β,n is a normalisation constant which can be explicitly computed as Selberg integral.

• eigenvalues repel as the joint probability density has a zero (of \(\beta\)th order) for coinciding eigenvalues \(\lambda_j=\lambda_i\).

computational resource

• https://drive.google.com/file/d/0B8XXo8Tve1cxZEdFLUVsM0hwUWM/view