Type: Article
Publication Date: 2012-09-12
Citations: 30
DOI: https://doi.org/10.1007/s00440-012-0450-3
We show that the distribution of (a suitable rescaling of) a single eigenvalue gap $$\lambda _{i+1}(M_n)-\lambda _i(M_n)$$ of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin–Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin–Mehta law required either an averaging in the eigenvalue index parameter $$i$$ , or fixing the energy level $$u$$ instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function $$N_{(-\infty ,x)}(\tilde{M}_n)$$ (where $$\tilde{M}_n$$ is a suitably rescaled version of $$M_n$$ ) with the event that there is no spectrum in an interval $$[x,x+s]$$ , in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.