On the empty balls of a critical super-Brownian motion

Let $\{X_t{\}}_{t\ge 0}$ be a $d$-dimensional critical super-Brownian motion started from a Poisson random measure whose intensity is the Lebesgue measure. Denote by $R_t:=\sup \{u>0:X_t(\{x\in {\mathbb{R}}^d:\mathrm{\mid }x\mathrm{\mid }<u\})=0\}$ the radius of the largest empty ball centered at the origin of $X_t$. In this work, we prove that for $r>0$, $$\underset{t\to \mathrm{\infty }}{\lim }\mathbb{P}(\frac{R_t}{t^{(1\mathrm{/}d)\wedge (3-d{)}^{+}}}\ge r)=e^{-A_d(r)},$$ where $A_d(r)$ satisfies ${\lim }_{r\to \mathrm{\infty }}\frac{A_d(r)}{r^{\mathrm{\mid }d-2\mathrm{\mid }+d{\text{ind}}_{\{d=2\}}}}=C$ for some $C\in (0,\mathrm{\infty })$ depending only on $d$.