Random Waves On $\mathbb T^3$: Nodal Area Variance and Lattice Point Correlations

Abstract

We consider the ensemble of random Gaussian Laplace eigenfunctions on T 3 = R 3 /Z 3 ('3d arithmetic random waves'), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or 'energy', of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.

Nodal area distribution for arithmetic random waves

“…X k,k (n) = 1 nN n λ 1 ,...,λ 4 ∈Xn(4) λ 1,(k) λ 2,(k) a λ 1 a λ 2 a λ 3 a λ 4 , X k,k,j,j (n) = 1 n 2 N n λ 1 ,...,λ 4 ∈Xn(4) λ 1,(k) λ 2,(k) λ 3,(j) λ 4,(j) a λ 1 a λ 2 a λ 3 a λ 4 , λ (k) denotes the k-th component of λ, and X n (4) is the set of d-dimensional lattice point, nondegenerate, 4-correlations defined in Section 3.3 below (see also [1,Section 1.4]).…”

“…In dimension d ≥ 5 the set of non-degenerate tuples X n (4) is much larger than D n (4) [1], as opposed to what happens in dimensions 2 and 3. This implies that the derivation of the asymptotic behavior of variance and limiting distribution of the nodal volume requires a precise analysis of the structure of the non-degenerate tuples X n (4) which seems to be very technically demanding.…”

“…where δ x is the Dirac delta centred at x ∈ S 1 . It was shown [16, Theorem 1.1] that, if {n i } i≥1 is any sequence of elements in S such that N n i → ∞, then 1] is the Fourier transform of µ n . The positive real numbers c n i in the leading constant depend on the angular distribution of Λ n i , i.e., in dimension d = 2, the asymptotic behavior of the variance is non-universal.…”

“…Nodal area. The asymptotic behavior of the nodal area variance on T 3 has been recently analysed in [1]: the variance of the nodal area has the following precise asymptotic behavior, as n → ∞, n ≡ 0, 4, 7 (mod 8), (1) .…”

Nodal Intersections for Arithmetic Random Waves Against a Surface

“…We work with an ensemble of random Gaussian Laplace toral eigenfunctions ('arithmetic random waves' [32,35,26]) (1) ,µ (2) ,µ (3) )∈E a µ e 2πi µ,x ,…”