Index Distribution of Gaussian Random Matrices
Journal Article
·
· Physical Review Letters
- Laboratoire de Physique Theorique et Modeles Statistiques (UMR 8626 du CNRS), Universite Paris-Sud, Batiment 100, 91405 Orsay Cedex (France)
- Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste (Italy)
We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N{sub +}) of a random NxN matrix belonging to Gaussian orthogonal (beta=1), unitary (beta=2) or symplectic (beta=4) ensembles. The distribution of the fraction of positive eigenvalues c=N{sub +}/N scales, for large N, as P(c,N){approx_equal}exp[-betaN{sup 2}PHI(c)] where the rate function PHI(c), symmetric around c=1/2 and universal (independent of beta), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic singularity in the rate function.
- OSTI ID:
- 21370863
- Journal Information:
- Physical Review Letters, Journal Name: Physical Review Letters Journal Issue: 22 Vol. 103; ISSN 0031-9007; ISSN PRLTAO
- Country of Publication:
- United States
- Language:
- English
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