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.