NEW SPECTRAL STATISTICS FOR ENSEMBLES OF 2 × 2 REAL SYMMETRIC RANDOM MATRICES

Sachin Kumar, Zafar Ahmed

Abstract


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.

Keywords


real symmetric matrices; Wigner surmise

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ISSN 1210-2709 (Print)
ISSN 1805-2363 (Online)
Published by the Czech Technical University in Prague