Indistinguishability of unbounded components in the occupied and vacant sets of Boolean models on symmetric spaces
Yingxin Mu, Artem Sapozhnikov
TL;DR
The paper analyzes Boolean models on non-compact symmetric spaces driven by insertion- or deletion-tolerant point processes and proves that, when there are infinitely many unbounded components, the unbounded components of the occupied set $\mathcal{O}$ and vacant set $\mathcal{V}$ are indistinguishable by any isometry-invariant property. It develops a continuum toolbox—mass-transport on symmetric spaces, pivotal balls, and a forest of trifurcations with a stationary environment for a random walk on the trifurcation forest—to establish indistinguishability and consequences such as monotonicity of uniqueness and connectivity decay. It also demonstrates percolation phase transitions within unbounded components and shows transience of infinite occupied components, with extensions to random radii models and hyperbolic-like settings. The work generalizes Lyons–Schramm’s discrete indistinguishability to a continuum framework, providing new insights into phase transitions, connectivity, and random-walk behavior on occupied and vacant sets in symmetric spaces.
Abstract
We study Boolean models on Riemannian symmetric spaces driven by homogeneous insertion- or deletion-tolerant point processes. We prove that in both the set covered by the balls (the occupied set) and its complement (the vacant set), one cannot distinguish unbounded components from each other by any isometry invariant component property. This implies the uniqueness monotonicity for the occupied and vacant sets of Poisson-Boolean models and an equivalence of non-uniqueness to the decay of connectivity for both sets. These results are continuum analogues of those by Lyons and Schramm arXiv:math/9811170. However, unlike the proof of the indistinguishability in arXiv:math/9811170, our proof does not rely on transience of unbounded components. We also prove the existence of a percolation phase transition for independent Poisson-Boolean model on unbounded connected components of both occupied and vacant sets and show transience of a random walk on the occupied set. Apart from some technical differences, we treat the occupied and the vacant sets of Boolean models within a single framework.