The Index Distribution of Gaussian Random Matrices

Details The Index Distribution of Gaussian Random Matrices The Index Distribution of Gaussian Random Matrices

2009-10-05

arxiv.org/abs/0910.0775v1

Phys. Rev. Lett. 103, 220603 (2009)

Physics

Condensed Matter

Statistical Mechanics

4 pages Revtex, 4 .eps figures included

Scientific paper

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N_{+}) 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_{+}/N scales, for large N, as Prob(c,N)\simeq\exp[-\beta N^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.

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