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2 added explicit formulas for the N-dependence

The tail of the eigenvalue density is indeed described by the TW distribution, simply because it is entirely due to the largest eigenvalue. (The probability that there are two eigenvalues outside of the hard upper bound is negligibly small.) The fact that the TW distribution holds only asymptotically should need not stop you. It simply means that the functional form which you use for the tail is only accurate if the dimensionality $N$ of the matrix is much larger than unity.

For a specific example of the use of the TW distribution to describe the tail of the GOE semicircle, see http://arxiv.org/abs/cond-mat/0006375 (Fig. 2 compares the tail of the eigenvalue density with the TW distribution.)

To be a bit more explicit about the $N$-dependence of the tail distribution, let's consider an $N\times N$ real symmetric matrix $H$ from the GOE, with mean level spacing $\delta$ (independent of $N\gg 1$) in the bulk of the spectrum. The semicircular energy level density is

$\rho(E)=(1/\delta)[1-(E/E_0)^2]^{1/2}$

for $|E|\lt E_0$ and zero for $|E|>E_0$. The hard upper bound is at $E_0=2N\delta/\pi$ (as required by the normalization $\int\rho(E)dE=N$).

The tail distribution for $E\gg E_0$ has the Tracy-Widom form

$\rho(x)=(4\sqrt{\pi}x^{1/4})^{-1}\exp[-(2/3)x^{3/2}]$

with $x=\pi N^{-1/3}(E-E_0)/\delta$, valid in the range $1\ll x\ll N^{2/3}$. This is a wide range for $N\gg 1$.

1

The tail of the eigenvalue density is indeed described by the TW distribution, simply because it is entirely due to the largest eigenvalue. (The probability that there are two eigenvalues outside of the hard upper bound is negligibly small.) The fact that the TW distribution holds only asymptotically should not stop you. It simply means that the functional form which you use for the tail is only accurate if the dimensionality of the matrix is much larger than unity.

For a specific example of the use of the TW distribution to describe the tail of the GOE semicircle, see http://arxiv.org/abs/cond-mat/0006375 (Fig. 2 compares the tail of the eigenvalue density with the TW distribution.)