Nodal intersections for random waves on the 3-dimensional torus

Abstract

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard three-dimensional flat torus with a fixed smooth reference curve, which has nowhere vanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result gives a bound for the variance, if either the torsion of the curve is nowhere zero or if the curve is planar. Date: September 25, 2018. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreements n o 320755 (Z.R.) and n o 335141 (I.W.), and from the Friends of the Institute for Advanced Study (Z.R.).1 By a curve we always mean a parameterized, compact, immersed curve.

Nodal Intersections for Arithmetic Random Waves Against a Surface

“…Throughout we apply many ideas of [35,26,36,37]. The arithmetic random wave F (d) : T d → R (1.6) is a random field.…”

“…The arithmetic random wave F (d) : T d → R (1.6) is a random field. The number of nodal intersections Z (d) (1.7) against a curve are the zeros of a process, which is the restriction of the random wave F to the curve [36,34,37]. For a smooth process p : T → R defined on an appropriate parameter set T ⊂ R, moments of the number of zeros may be computed, under certain assumptions, via Kac-Rice formulas [2,13].…”

“…The zero line of f is not necessarily (isometric to) the nodal intersection curve {x ∈ T 3 : F (x) = 0}: rather, it is isometric to the projection of the nodal intersection curve onto the domain U of the parametrisation γ. Therefore, the application of Kac-Rice formulas for f yields the moments of the length of the projected curve (see [2,Theorem 11.3]), not of the intersection curve itself: this is in marked contrast with what happens in the case of the nodal intersections number [36,34,37].…”

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

“…(see e.g. [5,Section 1] or [29,Section 4]). We shall require a bound for the number of lattice points on circles centred in Z 3 ,…”

Random Eigenfunctions on Flat Tori: Universality for the Number of Intersections

“…However, we are not aware of similar deterministic results regarding the intersection with a smooth curve as in T 2 . On the probabilistic side, Rudnick, Wigman and Yesha [24] have recently extended Theorem 1.5 to T 3 . Here, for λ 2 = 4π 2 m with m = 0, 4, 7 mod 8, let E λ be the collection of µ =…”