Sodin's moderate and large deviation conjecture for planar GAF zeros

Let $n(r)$ be the number of zeros of the planar Gaussian analytic function $\mathbf g$ in the disk $D(0,r)$. Sodin's deviation conjecture. As $r\to\infty$,

$$\genfrac{}{}{}{}{\log\log\left(\genfrac{}{}{}{}{1}{{\bf P}[|n(r)-r^2|>r^\alpha]}\right)}{\log r}\longrightarrow \begin{cases} 2\alpha-1,&\genfrac{}{}{}{}{1}{2}\leq\alpha\leq 1,\\ 3\alpha-2,&1\leq\alpha\leq 2,\\ 2\alpha,&2\leq\alpha. \end{cases}$$

This predicts the logarithmic scale of deviations of the zero count across moderate and very large deviation regimes; the source presents it as a conjecture inspired by physical results for Coulomb gases.