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 $x\to\infty$ with fixed intensity parameter $\alpha$, and $\alpha\to\infty$ with fixed $x$). We also obtain a new family of martingales built on the conditioned Brownian motion, which may be of independent interest.

Comment: 30 pages, 4 figures