How many eigenvalues of a Gaussian random matrix are positive?

Area of Science:

• Mathematics
• Physics
• Statistics

Background:

• Gaussian random matrices are fundamental in various scientific fields.
• Understanding eigenvalue distributions is crucial for analyzing complex systems.

Purpose of the Study:

• To determine the probability distribution of positive eigenvalues in large Gaussian random matrices.
• To derive the large deviation rate function for the fraction of positive eigenvalues.
• To analyze the fluctuations of the index of positive eigenvalues.

Main Methods:

• Analytical calculations for large N (matrix dimension).
• Derivation of the large deviation rate function Φ(c).
• Analysis of the variance of index fluctuations.

Main Results:

• The index distribution follows P(N(+)=cN,N)~exp[-βN(2)Φ(c)] for large N.
• The rate function Φ(c) is explicitly computed and is independent of the Dyson index β.
• The variance of index fluctuations grows as Δ(N)~lnN/βπ(2) for large N.

Conclusions:

• The distribution of positive eigenvalues in large Gaussian random matrices can be precisely characterized.
• The derived formulas provide insights into the statistical properties of random matrices.
• Results are validated against exact formulas for specific cases (β=2).