Non-universality of the Nazarov–Sodin constant

On the Number of Nodal Domains of Toral Eigenfunctions

“…Moreover, there exist [7,Proposition 3.2] sequences {E} ⊆ S of energy levels with only o(E) nodal domains as E → ∞ for "most" coefficients (a ξ ) ξ∈E . However, it is widely believed that such a situation is impossible in some generic scenario, and it is desirable to show a lower bound of the "correct" order of magnitude N fa E (1.8) holding uniformly for generic sequences of energies {E} and coefficients (a ξ ) satisfying some mild extra assumptions 1 ; (1.8) will follow as a straightforward corollary of our main results (cf.…”

“…More generally, if |a ξ | is constant for ξ ∈ {(±ξ 1 , ±ξ 2 ), (±ξ 2 , ±ξ 1 )} then μ E,a ∈ P 0 (S 1 ). By [7,Theorem 2.1] there are exactly two measures in P 0 (S 1 ) with vanishing Nazarov-Sodin constant, the Cilleruelo measure…”

“…Choosing T 0 large (depending on 7 and η), and taking into account our choice of Υ, we see that the right-hand side of (6.44) is at most 7 3 , and this is uniform in β.…”

“…Let 0 < 7 , δ, η < 1 and K, L > 1 be given. Suppose that N → ∞ as E → ∞ and that the coefficients a ξ are of class A(g), for some slowly growing function g. There exists λ ∈ Λ 6 7 ,η , B 2 = B 2 (K, L, 7 , η) and E 8 = E 8 (K, L,7 , δ, η, γ, g) such that if E satisfies I(γ, B 2 ) for all E and for some fixed γ, then for all E ≥ E 8 |P[c ∈ Q(λ, L)] − P[b ∈ Q(λ, L)]| ≤ 7 (6.38) and P[c ∈ Q λ,L,η β ] − P[b ∈ Q λ,L,η β ] ≤ 7 (6.39) for all β.…”

“…It was conjectured[7, Conjecture 3.4] that c 1 = c 0 is the universal constant from (1.7), also known to be valid for Berry's Random Wave Model (equivalently random spherical harmonics[9]). …”