Mikhail SODIN

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Notation: N(fL) the number of components of the zero set Z(fL)
Theorem II: Suppose that (fL) is a C 2 -smooth Gaussian ensemble on
X, which has translation-invariant local limits a.e. on X. Suppose that
the local limiting spectral measures ρx have no atoms and satisfy the
non-degeneracy condition from the previous slide.
Then the function x ↦→ ν(ρx) belongs to L∞ (X), and
lim
L→∞ E
L −m
N(fL) − ν(ρx)dvol(x) = 0 .
Here, ν(ρx) is a limiting constant from Theorem I (Euclidean case).
Remark: One can see that
X ν(ρx) dvol(x) does not depend on the choice
of the Riemannian metrics, only the smooth structure on X matters.
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X
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