The inradius law for the cavities of planar Brownian motion, with applications to the Wiener sausage

Abstract

Let $K_1=B[0,1]$ be the trace of a planar Brownian motion, and let $N_{\mathrm{in}}(\eta )$ count the bounded components of ${\mathbb{R}}^2\setminus K_1$ with inradius greater than $\eta$. As $\eta \downarrow 0$, we prove that ${\eta }^2(\log (1/\eta ){)}^2{N}_{\mathrm{in}}(\eta )\to c_{\mathrm{in}}=(\pi /2)\int G^{2}d\mathcal{L}\in (0,1/2]$ almost surely and in $L^p$ for every $p\in [1,2]$, where $G$ is the inradius divided by the square root of the area and $\mathcal{L}$ is Werner's limiting shape law. Inradius determines survival under thickening: a cavity contains a hole of the Wiener sausage of radius $r$ if and only if its inradius exceeds $r$. Brownian scaling shows that, for fixed $r>0$ as $T\to \infty$, the number of cavities of $B[0,T]$ with inradius greater than $r$ is asymptotic to $4c_{\mathrm{in}}T/(r^2(\log T{)}^2)$ in probability and in $L^p$, $p\in [1,2]$. It also gives lower bounds for the sausage's hole count, the Betti curve of its degree-one persistent homology, and its radius integrals, with constant $4c_{\mathrm{in}}$ in expectation and any smaller constant with probability tending to one. A matching upper bound cannot follow from general bounds in terms of perimeter, area, diameter or length. We prove that it is equivalent to a first-moment estimate for near-critical pockets: holes at level $\eta$ on which the distance to the trace is at most $2\eta$. Equivalently, it suffices to control residual pockets beyond one per cavity with inradius in $(\eta ,2\eta ]$. This estimate is proved in a companion paper, yielding the exact order $T/(r^2(\log T{)}^2)$ for the expected number of holes.

arXiv commentThis version supersedes the former version v1 (posted under the title "Persistent homology of the planar Wiener sausage: Brownian scaling and a logarithmic expectation law"). It corrects errors in v1 and substantially revises it