The inradius law for the cavities of planar Brownian motion, with applications to the Wiener sausage
Abstract
Let K1=B[0,1] be the trace of a planar Brownian motion, and let Nin(η) count the bounded components of ℝ2∖ K1 with inradius greater than η. As η↓0, we prove that η2(log(1/η))2Nin(η)→ cin=(π/2)∫ G2dℒ∈(0,1/2] almost surely and in Lp for every p∈[1,2], where G is the inradius divided by the square root of the area and 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→∞, the number of cavities of B[0,T] with inradius greater than r is asymptotic to 4cinT/(r2(log T)2) in probability and in Lp, p∈[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 4cin 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/(r2(log T)2) for the expected number of holes.
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BibTeX
@misc{guillaume2026inradius,
title = {{The inradius law for the cavities of planar Brownian motion, with applications to the Wiener sausage}},
author = {Tristan Guillaume},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.11248},
eprint = {2606.11248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11248},
}
RIS
TY - UNPB
TI - The inradius law for the cavities of planar Brownian motion, with applications to the Wiener sausage
AU - Tristan Guillaume
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2606.11248
UR - https://arxiv.org/abs/2606.11248
AB - Let K_1=B[0,1] be the trace of a planar Brownian motion, and let N_in(η) count the bounded components of ℝ^2∖ K_1 with inradius greater than η. As η↓0, we prove that η^2(log(1/η))^2N_in(η)→ c_in=(π/2)∫ G^2dℒ∈(0,1/2] almost surely and in L^p for every p∈[1,2], where G is the inradius divided by the square root of the area and 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→∞, the number of cavities of B[0,T] with inradius greater than r is asymptotic to 4c_inT/(r^2(log T)^2) in probability and in L^p, p∈[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_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.
ER -