The asymptotic distribution of a single eigenvalue gap of a Wigner matrix
Abstract
We show that the distribution of (a suitable rescaling of) a single eigenvalue gap
λ
i+1
(
M
n
)−
λ
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
(−∞,x)
(
M
~
n
)
(where
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.