Weak moderate deviations for the halo volume

Let $R$ satisfy $R\in(2,\frac12L)$, and let $B_R$ be the disc of radius $R$, with halo volume $V(\gamma)$ under the probability measure $\mu_\beta$. Write $I(B_R)$ for the halo rate function evaluated at $B_R$. There exists a function $\Psi_R\colon\mathbb{R}\to\mathbb{R}$ such that

$$\limsup_{\beta\to\infty}\frac{1}{\beta^{1/3}}\log\left\{\mathrm{e}^{\beta I(B_R)}\mu_\beta\left(\beta^{2/3}\left[V(\gamma)-\pi R^2\right]\in K\right)\right\}\leq\sup_K\Psi_R$$

for every compact $K\subset\mathbb{R}$, and

$$\liminf_{\beta\to\infty}\frac{1}{\beta^{1/3}}\log\left\{\mathrm{e}^{\beta I(B_R)}\mu_\beta\left(\beta^{2/3}\left[V(\gamma)-\pi R^2\right]\in O\right)\right\}\geq\sup_O\Psi_R$$

for every open $O\subset\mathbb{R}$. This weak moderate deviation principle refines the large deviation principle near a droplet of radius $R$ by describing fluctuations of the halo volume on the $\beta^{-2/3}$ scale after removal of the leading exponential cost.