Mikhail SODIN

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Translation-invariant local limits:
Definition: The Gaussian ensemble (fL) has translation-invariant local
limits as L → ∞ if for a.e. x ∈ X, there exists a Hermitean positive
definite function kx : R m → R 1 , such that for each R < ∞,
✫
lim
L→∞
sup
|u|,|v|≤R
|Kx,L(u, v) − kx(u − v)| = 0 .
The limiting kernels kx(u − v) are covariance kernels of
translation-invariant Gaussian functions Fx : R m → R 1 (x ∈ X).
They are the Fourier transforms of probability measures ρx on R m ,
symmetric w.r.t. the origin.
We call the function Fx the local limiting function, and the measure ρx
the local limiting spectral measure of the family fL at the point x.
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