On the variance of the nodal volume of arithmetic random waves

Abstract

Rudnick and Wigman (Ann. Henri Poincaré, 2008) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the d-dimensional torus is O(E/N ), as E → ∞, where E is the energy and N is the dimension of the eigenspace corresponding to E. Previous results have established this with stronger asymptotics when d = 2 and d = 3. In this brief note we prove an upper bound of the form O(E/N 1+α(d)−ǫ ), for any ǫ > 0 and d ≥ 4, where α(d) is positive and tends to zero with d. The power saving is the best possible with the current method (up to ǫ) when d ≥ 5 due to the proof of the ℓ 2 -decoupling conjecture by Bourgain and Demeter.

On the limiting behaviour of arithmetic toral eigenfunctions

“…In an effort to understand the variation of this quantity within these constraints, several authors have studied the behaviour of F −1 m (0) where F m is a random element of E m (see for instance [4], [39], [30], [35], [3], [10], [16]). The two most popular probability measures, chosen because they respect the symmetries of the model, are defined as follows.…”