Random matrices: why to distingusish bulk and edge cases?

Question

I have been recently reading some papers on universality of spectral statistics of random matrices written by Terry Tao, Van Vu, L. Erdos, H.T. Yau and others, and I am puzzled by such a dichotomy, namely:

one should treat the bulk case and edge case separately (using possibly different techniques).

Could someone please provide an intuition on why such a dichotomy is needed, e.g., in local semicircle law for Wigner matrices? (I naively thought that on the edge the limiting density takes value zero when it is compactly supported, which may cause singularity in certain resolvent; or near the edge, the density, if differentiable, displays a different rate of change.)

I would appreciate any thoughts and comments on this phenomenon. Thank you.

Regards,

Chee

Answer 1

Intuitively, the fact that the density vanishes at the edge already hints that the spacing at the edge changes (which indeed it does), and that the asymptotics are different; this is confirmed in the Gaussian setup - the different scaling leads to a different scaling of the orthogonal polynomials (Hermite in this case) and these asymptotics eventually lead to the Airy kernel for the correlation function instead of the Sine kernel.

On a more technical level, and relevant to universality, the map $z\to S(z)$, where $S(z)$ is the Stieltjes transform of the semi circle, has a singularity at the edge of the spectrum which requires care when controlling $S(z)$ with $z=E+i\eta$, $\eta$ small and $E$ at the edge.

At the end, the techniques are not that different, but the edge does require more refined analysis.

Answer 2

there is a mathematics reason to distinguish bulk from edge, as mentioned in Ofer Zeitouni's answer, and there is a physics reason: in a physical system with an excitation gap, for example, a superconductor, you might want to know the sample-to-sample fluctuations of the size of the gap; these are described by the Tracy-Widom edge statistics; on the other hand, the Wigner-Dyson bulk statistics describes the fluctuations of the level density far from the gap.