Index distribution of gaussian random matrices

Area of Science:

• Random Matrix Theory
• Quantum Chaos
• Statistical Physics

Background:

• Understanding eigenvalue distributions is crucial in various scientific fields.
• Gaussian ensembles (GOE, GUE, GSE) are fundamental models in random matrix theory.
• The behavior of eigenvalues in large random matrices reveals universal properties.

Purpose of the Study:

• To analytically compute the probability distribution of positive eigenvalues for large random matrices.
• To determine the scaling behavior and the universal rate function governing this distribution.
• To investigate the nature of the distribution's tails and peak.

Main Methods:

• Analytical computation for large N matrices.
• Focus on Gaussian orthogonal (beta=1), unitary (beta=2), and symplectic (beta=4) ensembles.
• Derivation of the scaling law P(c,N) approximately = exp[-betaN(2)Phi(c)].

Main Results:

• The distribution of the fraction of positive eigenvalues (c=N+/N) was determined.
• A universal, beta-independent rate function Phi(c) was exactly calculated.
• The distribution exhibits non-Gaussian tails and a logarithmic singularity at its peak.

Conclusions:

• The fraction of positive eigenvalues in large random matrices follows a universal scaling law.
• The rate function Phi(c) is independent of the matrix ensemble (beta).
• Deviations from Gaussian behavior are present even near the distribution's peak.