Index distribution of gaussian random matrices.

@article{Majumdar2009IndexDO,
  title={Index distribution of gaussian random matrices.},
  author={Satya N. Majumdar and C. Riera Nadal and Antonello Scardicchio and Pierpaolo Vivo},
  journal={Physical review letters},
  year={2009},
  volume={103 22},
  pages={
          220603
        }
}
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… CONTINUE READING

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