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We consider a wide class of families $(F_m)_{m\in\mathbb{N}}$ of Gaussianfields on $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ defined by \[F_m:x\mapsto\frac{1}{\sqrt{|\Lambda_m|}}\sum_{\lambda\in\Lambda_m}\zeta_\lambda e^{2\pii\langle \lambda,x\rangle}\] where the $\zeta_\lambda$'s are independent std.normals and $\Lambda_m$ is the set of solutions $\lambda\in\mathbb{Z}^d$ to$p(\lambda)=m$ for a fixed elliptic polynomial $p$ with integer coefficients.The case $p(x)=x_1^2+\dots+x_d^2$ is a random Laplace eigenfunction whose lawis sometimes called the $\textit{arithmetic random wave}$, studied in the pastby many authors. In contrast, we consider three classes of polynomials $p$: acertain family of positive definite quadratic forms in two variables, allpositive definite quadratic forms in three variables except multiples of$x_1^2+x_2^2+x_3^2$, and a wide family of polynomials in many variables.  For these classes of polynomials, we study the $(d-1)$-dimensional volume$\mathcal{V}_m$ of the zero set of $F_m$. We compute the asymptotics, as$m\to+\infty$ along certain sequences of integers, of the expectation andvariance of $\mathcal{V}_m$. Moreover, we prove that in the same limit,$\frac{\mathcal{V}_m-\mathbb{E}[\mathcal{V}_m]}{\sqrt{\text{Var}(\mathcal{V}_m)}}$converges to a std. normal.  As in previous works, one reduces the problem of these asymptotics to thestudy of certain arithmetic properties of the sets of solutions to$p(\lambda)=m$. We need to study the number of such solutions for fixed $m$,the number of quadruples of solutions $(\lambda,\mu,\nu,\iota)$ satisfying$\lambda+\mu+\nu+\iota=0$, ($4$-correlations), and the rate of convergence ofthe counting measure of $\Lambda_m$ towards a certain limiting measure on thehypersurface $\{p(x)=1\}$. To this end, we use prior results on this topic butalso prove a new estimate on correlations, of independent interest.

On the limiting behaviour of arithmetic toral eigenfunctions