New spectral statistics for ensembles of 2 × 2 real symmetric random matrices

Anotace: We investigate spacing statistics for ensembles of various real random matrices where the matrix-elements have various Probability Distribution Function (PDF: f(x)) including Gaussian. For two modifications of 2 × 2 matrices with various PDFs, we derive the spacing distributions p(s) of adjacent energy eigenvalues. Nevertheless, they show the linear level repulsion near s = 0 as αs where α depends on the choice of the PDF. More interestingly when f(x) = xe−x2 (f(0) = 0), we get cubic level repulsion near s = 0: p(s) ~ s3e−s2.We also derive the distribution of eigenvalues D(ε) for these matrices.