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Extreme value statistics of eigenvalues of Gaussian random matrices
David S Dean1, Satya N Majumdar
1Laboratoire de Physique Théorique UMR 5152 du CNRS, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex 4, France.
Summary
We derived exact probabilities for extreme eigenvalues in random matrices. For large matrices, the likelihood of all eigenvalues being positive or negative decays exponentially with N-squared, featuring a universal exponent.
Area of Science:
- * Mathematical Physics
- * Condensed Matter Physics
- * Statistical Mechanics
Background:
- * Random matrix theory (RMT) is crucial for understanding complex systems.
- * Gaussian ensembles (GOE, GUE, GSE) are fundamental models in RMT.
- * Large deviation theory analyzes rare events in probability distributions.
Purpose of the Study:
- * To compute exact asymptotic results for large deviations of extreme eigenvalues in Gaussian ensembles.
- * To determine the probability of all eigenvalues being positive or negative.
- * To generalize the Wigner semi-circle law for restricted eigenvalue distributions.
Main Methods:
- * Analytical computation of exact asymptotic formulas.
- * Application of large deviation techniques to eigenvalue statistics.
- * Calculation of joint probability distributions for minimum and maximum eigenvalues.
Main Results:
- * Probability of all eigenvalues being positive/negative decays as exp[-beta * theta(0) * N^2].
- * Universal exponent theta(0) = (ln 3)/4 found.
- * Exact joint probability distribution for min/max eigenvalues derived.
- * Generalized Wigner semi-circle law for restricted ensembles, showing inverse square-root singularities.
Conclusions:
- * The study provides precise analytical results for extreme eigenvalue statistics in RMT.
- * Findings confirm the universality of certain statistical properties in random matrices.
- * The generalized Wigner law offers new insights into restricted eigenvalue systems.
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