Insertion and deletion tolerance of point processes (Q388960)

Let \(\Pi\) be a simple point process in \({\mathbb R}^d\). Let \(\prec\) denote the absolute continuity in law, i.e., for random variables \(X\) and \(Y\) taking values in the same measurable space, \(X\prec Y\) if and only if \(\operatorname P(Y\in A)=0\) implies \(\operatorname P(X\in A)=0\) for all measurable \(A\). Let \({\mathcal B}^d\) denote the Borel \(\sigma\)-algebra in \({\mathbb R}^d\) and \(L\) be the Lebesgue measure. \(\Pi\) is insertion-tolerant if for every \(B\in {\mathcal B}^d\) with \(L(B)\in (0, \infty)\) holds: if \(U\) is uniformly distributed in \(B\) and independent of \(\Pi\), then \(\Pi + \delta_U\prec \Pi\), where \(\delta_x\) denotes the point measure at \(x\in {\mathbb R}^d\). Let \({\mathbf M}\) denote the space of simple point measures in \({\mathbb R}^d\). The support of a measure \(\mu\in {\mathbf M}\) is denoted by \([\mu] =\{y\in {\mathbb R}^d:\,\, \mu(\{y\})=1\}\). A \(\Pi\)-point is an \({\mathbb R}^d\)-valued variable \(Z\) such that \(Z\in [\Pi]\) a.s. A finite subprocess of \(\Pi\) is a point process \({\mathcal F}\) such that \({\mathcal F}({\mathbb R}^d)< \infty\) and \([{\mathcal F}] \subset [\Pi]\) a.s. The point process \(\Pi\) is called deletion-tolerant if for any \(\Pi\)-point \(Z\) we have \(\Pi - \delta_Z\prec \Pi\). The authors prove several equivalent formulations of insertion-tolerance and deletion-tolerance conditions. A point process is translation-invariant if it is invariant in law under the group of all translations of \({\mathbb R}^d\).NEWLINENEWLINE Theorem 1. A translation-invariant point process \(\Pi\) in \({\mathbb R}^d\) is insertion-tolerant if and only if there exists \(B\in {\mathcal B}^d\) with \(L(B)\in (0, \infty)\) such that holds: if \(U\) is uniformly distributed in \(B\) and independent of \(\Pi\), then \(\Pi + \delta_U\prec \Pi\).NEWLINENEWLINE The Boolean continuum percolation model for point processes is defined as follows( see [\textit{R. Meester} and \textit{R. Roy}, Continuum percolation. Cambridge: Cambridge Univ. Press (1996; Zbl 0858.60092)]): For \(R>0\) and \(\mu\in {\mathbf M}\), consider the set \(O(\mu)= \cup_{x\in[\mu]} B(x, R)\), where \(B(x, R)\) is the open ball of radius \(R\) with center \(x\). \(O(\mu)\) is called the occupied region. The connected components of \(O(\mu)\) are called clusters.NEWLINENEWLINE Theorem 2 (continuum percolation). Let \(\Pi\) be a translation-invariant ergodic insertion-tolerant point process in \({\mathbb R}^d\). For any \(R>0\), the occupied region \(O(\Pi)\) has at most one unbounded cluster a.s.NEWLINENEWLINE The proof of Theorem 2 is similar to the uniqueness proofs in Chapter 7.