On the limiting behaviour of arithmetic toral eigenfunctions

where the ξ λ 's are independent standard normals and Λ m is the set of solutions λ∈ℤ d to the equation p(λ)=m for some fixed elliptic polynomial p with integer coefficients. The case p(x)=x 1 2 +⋯+x d 2 amounts to considering a random Laplace eigenfunction whose law is sometimes called the arithmetic random wave and has been studied in the past by many authors. In contrast, we consider three classes of polynomials p: a certain family of positive definite quadratic forms in two variables, all positive definite quadratic forms in three variables except the multiples of x 1 2 +x 2 2 +x 3 2 , and a wide family of polynomials in many variables.