Mikhail SODIN

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Step 3: The Kac-Rice bound for the number of critical pts:
Lemma: EN(r; F ) volm−1∂B(r).
=⇒ a.s., lim R
✫
N(R; F )
volB(R) exists and equals lim r
EN(r; F )
volB(r)
=: ν(ρ)
Step 4: Positivity of ν(ρ): We need to show: for some r0 > 0, EN(r0; F ) > 0.
Standard Gaussian Lemma: Suppose µ is a compactly supported measure
with spt(µ) ⊂ spt(ρ). Then for each ball B ⊂ R m and each ɛ > 0,
P F − µ C( ¯ B) < ɛ > 0.
By assumption (∗) in Theorem I, ∃ such a measure µ with Z(µ) having a
bounded connected component.
By real analyticity of µ, this component is isolated.
Choosing ɛ small enough, we get P N(r0; F ) > 0 > 0 for some r0
=⇒ EN(r0; F ) > 0.
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