Index distribution of gaussian random matrices.

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N+) of a random N x N 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+/N scales, for large N, as P(c,N) approximately = exp[-betaN(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.

• Publication year

  2009

• Venue

  Physical Review Letters

• Publication date

  2009-10-05

• Fields of study

  Mathematics, Physics, Medicine

• Identifiers
  DOI 10.1103/PhysRevLett.103.220603arXiv 0910.0775PMID 20366083
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar, PubMed

• No claims are published for this paper.

• No concepts are published for this paper.

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