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

TL;DR

This paper investigates the shape of the vacant regions in two-dimensional Brownian interlacements, showing they resemble a scaled Brownian amoeba near trajectories, and introduces new martingales related to conditioned Brownian motion.

Contribution

It characterizes the limiting shape of connected components of the vacant set and introduces a novel family of martingales for conditioned Brownian motion.

Findings

Connected components resemble scaled Brownian amoebas near trajectories.

Established distributional closeness of components to Brownian amoebas.

Developed new martingales for conditioned Brownian motion.

Abstract

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.