Maximal Nazarov–Sodin constant conjecture for random plane waves and arithmetic random waves

Let $\mathcal{P}_{\text{symm}}$ be the class of symmetric spectral probability measures, and let $\{\mu_n\}$ be the sequence of arithmetic random wave spectral measures. Write $c_{\mathrm{NS}}(\rho)$ for the Nazarov–Sodin constant associated with a spectral measure $\rho$, and let $c_{\mathrm{RWM}}$ denote the constant for the uniform measure on $\mathcal{S}^{1}$. Let $d_{\max}$ and $c_{\max}$ be the maximal values of the Nazarov–Sodin constant in the arithmetic random wave and general spectral-measure settings, respectively.

Maximal Nazarov–Sodin constant conjecture. For every $\mu\in\mathcal{P}_{\text{symm}}$ that is a weak-$*$ limit of $\{\mu_n\}$, the maximal value $d_{\max}$ is uniquely attained by $c_{\mathrm{NS}}(\mu_{\mathcal{S}^{1}})$, where $\mu_{\mathcal{S}^{1}}$ is the uniform measure on $\mathcal{S}^{1}\subseteq\mathbb{R}^{2}$. In particular,

$$d_{\max}=c_{\mathrm{RWM}}.$$

For $\rho\in\mathcal{P}$, the maximal value $c_{\max}$ is uniquely attained by $c_{\mathrm{NS}}(\rho)$ when $\rho$ is the uniform measure on $\mathcal{S}^{1}\subseteq\mathbb{R}^{2}$. In particular,

$$c_{\max}=d_{\max}=c_{\mathrm{RWM}}.$$

The conjecture proposes that the uniform spectral measure maximizes the Nazarov–Sodin constant both among weak-$*$ limits of arithmetic random wave spectral measures satisfying the stated symmetry condition and among all admissible spectral measures. The surrounding discussion gives motivation from comparisons with concentrated measures such as Cilleruelo measures, but no resolution is supplied here.