On the shape of the connected components of the complement of two-dimensional Brownian random interlacements

We study the limiting shape of the connected components of the vacant set of two-dimensional Brownian random interlacements: we prove that the connected component around $x$ is close in distribution to a rescaled \emph{Brownian amoeba} in the regime when the distance from $x\in \mathbb{C}$ to the closest trajectory is small (which, in particular, includes the cases $x\to \mathrm{\infty }$ with fixed intensity parameter $\alpha$, and $\alpha \to \mathrm{\infty }$ with fixed $x$). We also obtain a new family of martingales built on the conditioned Brownian motion, which may be of independent interest.