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

TL;DR

This work analyzes the vacant set of two-dimensional Brownian random interlacements and characterizes the limiting shape of its connected components. The authors show that, when the distance to the nearest trajectory is small, a component around a point is well approximated by a rescaled Brownian amoeba, leveraging Poissonian decompositions of trajectories and capacity-based estimates. They also introduce a novel family of martingales for conditioned Brownian motion, enabling precise hitting-probability control, and study the central-cell geometry in the high-intensity limit. Together, these results provide a detailed geometric description of the vacancy structure and expand the toolkit for studying conditioned Brownian motion in random interlacement models.

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

Figures (4)

• Figure 1: On the equivalent definition of $\mathop{\mathrm{BRI}}(\alpha)$.
• Figure 2: The Brownian amoeba ${\mathfrak A}$
• Figure 3: The central cell (in this case, formed by five bi-infinite trajectories).
• Figure 4: On the definition of the event $U'_r$

Theorems & Definitions (28)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Theorem 2.4
• Definition 2.5
• Theorem 2.6
• Theorem 2.7
• Lemma 3.1
• proof