Let \(\{X_t\}_{t\ge 0}\) be a d-dimensional critical or subcritical super-Brownian motion started from the Lebesgue measure. Denote by \(R_t:=\sup \{u>0: X_t(\{x\in \mathbb {R}^d:|x|< u\})=0\}\) the radius of the largest empty ball centred at the origin of \(X_t\) . This article shows that \(R_t\) will converge in law after suitable renormalization as \(t\rightarrow \infty \) . Our results indicate that the renormalization scale of \(R_t\) not only depends on the dimension but also depends on whether the branching mechanism has finite variance and is critical or not, which completes our previous results for the critical super-Brownian with quadratic branching mechanism (Zhang and Xiong in Bernoulli 30:72–87, 2024) and reveals some new phase transitions.
