The inradius law for the cavities of planar Brownian motion, with applications to the Wiener sausage
Year: 2026
Type: preprint
Abstract: Let $K_1=B[0,1]$ be the trace of a planar Brownian motion, and let $N_{\mathrm{in}}(η)$ count the bounded components of $\mathbb{R}^2\setminus K_1$ with inradius greater than $η$. As $η\downarrow0$, we prove that $η^2(\log(1/η))^2N_{\mathrm{in}}(η)\to c_{\mathrm{in}}=(π/2)\int G^2d\mathcal{L}\in(0,1/2]$ 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 $η$ on which the distance to the trace is at most $2η$. Equivalently, it suffices to control residual pockets beyond one per cavity with inradius in $(η,2η]$. This estimate is proved in a companion paper, yielding the exact order $T/(r^2(\log T)^2)$ for the expected number of holes.
Source: arXiv (Cornell University)
Author: Guillaume, Tristan
Institution: CY Cergy Paris Université
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164369388