Orphée Collin ; Serguei Popov - On the shape of the connected components of the complement of two-dimensional Brownian random interlacements

cm:14455 - Communications in Mathematics, March 3, 2025, Volume 33 (2025), Issue 1 - https://doi.org/10.46298/cm.14455
On the shape of the connected components of the complement of two-dimensional Brownian random interlacementsArticle

Authors: Orphée Collin ; Serguei Popov

    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.


    Volume: Volume 33 (2025), Issue 1
    Published on: March 3, 2025
    Accepted on: January 30, 2025
    Submitted on: October 15, 2024
    Keywords: Mathematics - Probability,60K35, 60J65

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