Large Deviations of Extreme Eigenvalues of Random Matrices

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an ($N\times N$) random matrix are positive (negative) decreases for large $N$ as $\sim \exp [-\beta \theta (0)N^2]$ where the parameter $\beta$ characterizes the ensemble and the exponent $\theta (0)=(\ln 3)/4=0.274653\dots$ is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number $\zeta$, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at $\zeta$.

Figure 1

The dashed line shows the semicircular form of the average density of states. The largest eigenvalue is centered around its mean $\sqrt{2N}$ and fluctuates over a scale of width $N^{-1/6}$. The probability of fluctuations on this scale is described by the Tracy-Widom distribution (shown schematically).

Figure 2

The average density of states $f(x)$ plotted as a function of the shifted variable $x$ for $z=-1$ (dotted line), $z=0$ (solid line), and $z=0.5$ (dashed line).

Figure 3

Monte Carlo computation of $\ln [Q_N(0)]$ points with error bars along with a quadratic fit (solid line).

Figure 4

The analytic large $N$ formula for $f$ with $z=0$ (solid line) in Eq. (15) is compared to the numerically generated averaged histogram of ($6\times 6$) Gaussian matrices with positive eigenvalues. Despite the small size $N=6$, the agreement is already fairly good, except near the large $\mu$ tail.