# Does the Tracy-Widom distribution describe the tails of eigenvalue densities of finite dimensional random matrices?

The Tracy-Widom distribution (TW) describes the density of the largest eigenvalue of a random Hermitian matrix, when scaled and centered appropriately (depending on GOE/GUE/GSE/Wishart, etc).

In a recent manuscript of mine (non-math journal), I had casually mentioned that in finite dimensional, random Hermitian matrices, the tail that extends outside the hard upperbounds in the asymptotic densities (semicircle for GOE/GUE and Marchenko-Pastur for real/complex Wishart) is also described by TW. What I had intended was that the tail is formed by the largest eigenvalue and the density of this is still governed by TW.

However, the editor of the journal commented that TW is applicable only in the asymptotic limit. Is this true? In Johnstone's 2001 paper "On the distribution of the largest eigenvalue in principal component analysis", he derives a set of scaling constants for the largest eigenvalue of real Wishart matrices that explicitly depend on the dimensionality of the matrix.

Is there a disconnect between my understanding, what the editor said and what I've read in Johnstone's paper?

-
The Tracy-Widom law is indeed only valid asymptotically, even after rescaling. (Note for instance the arrow, rather than the equivalence symbol, in Theorem 1 of Johnstone's paper.) –  Terry Tao Dec 15 '11 at 5:24

As Terry pointed out in a comment, the TW law is only valid asymptotically. However, there are various results which show that the asymptotics for finite dimension, which TW heuristically suggest, are valid. This is analogous to the fact that the classical central limit theorem tells you nothing about the tails of a sum of independent random variables, but there exist other results which say that, for certain random variables, the tail estimates you'd guess from CLT are correct.

The first such result I know of for TW-type tails is due to Guillaume Aubrun, see this page; as Aubrun points out there, similar results have been proved by Michel Ledoux as well.

-

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 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 $N\gg 1$.

-