[Paper Review] Insertion and Deletion Tolerance of Point Processes

Alexander E. Holroyd, Terry Soo|Warwick Research Archive Portal (University of Warwick)|Jul 20, 2010

Stochastic processes and statistical mechanics11 references4 citations

TL;DR

This paper introduces and analyzes insertion and deletion tolerance for point processes on R^d, establishing equivalent characterizations using absolute continuity of laws under random point insertions or deletions. Key results show that translation-invariant processes like Gaussian zero processes on the hyperbolic plane are both insertion- and deletion-tolerant, with applications to continuum percolation and stable matching.

ABSTRACT

We develop a theory of insertion and deletion tolerance for point processes. A process is insertion-tolerant if adding a suitably chosen random point results in a point process that is absolutely continuous in law with respect to the original process. This condition and the related notion of deletion-tolerance are extensions of the so-called finite energy condition for discrete random processes. We prove several equivalent formulations of each condition, including versions involving Palm processes. Certain other seemingly natural variants of the conditions turn out not to be equivalent. We illustrate the concepts in the context of a number of examples, including Gaussian zero processes and randomly perturbed lattices, and we provide applications to continuum percolation and stable matching.

Motivation & Objective

• To formalize and analyze insertion and deletion tolerance as probabilistic invariance properties for point processes.
• To establish equivalent characterizations of insertion and deletion tolerance using absolute continuity of laws under random point operations.
• To investigate the implications of these conditions in the context of translation-invariant and ergodic point processes.
• To apply the theory to concrete models such as Gaussian zero processes and randomly perturbed lattices.
• To demonstrate applications in continuum percolation and stable matching via the tolerance properties.

Proposed method

• Define insertion-tolerance as the condition that adding a uniformly random point results in a law absolutely continuous with respect to the original process.
• Define deletion-tolerance as the condition that removing a point at a typical location results in a law absolutely continuous with respect to the original process.
• Prove multiple equivalent formulations of both tolerance conditions, including versions involving Palm processes and restrictions to bounded sets.
• Use the theory of Palm measures and ergodicity to derive characterizations for translation-invariant processes.
• Apply Rouché’s theorem and convergence of conditional laws in complex analysis to establish absolute continuity for Gaussian zero processes on the hyperbolic plane.
• Use diagonalization and martingale convergence arguments to prove almost sure convergence of conditional probabilities in the context of point process limits.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a point process to be insertion-tolerant or deletion-tolerant?
• RQ2 How do insertion and deletion tolerance relate to the finite energy condition and Palm measures?
• RQ3 Can the tolerance conditions be characterized using restrictions to bounded sets or conditional laws?
• RQ4 Do the natural analogues of the main theorems hold for deletion-tolerance as they do for insertion-tolerance?
• RQ5 Are Gaussian zero processes on the hyperbolic plane insertion- and deletion-tolerant, and what does this imply for their probabilistic structure?

Key findings

• Insertion-tolerance is equivalent to the absolute continuity of the law of the process with a random finite number of points inserted, under conditions on the insertion locations.
• Deletion-tolerance is equivalent to the almost sure positivity of the probability that a bounded set contains no points, given the process outside that set.
• For translation-invariant ergodic point processes, insertion-tolerance is equivalent to the absolute continuity of the Palm version with respect to the process with a point added at the origin.
• Deletion-tolerance does not imply the natural analogue of the Palm condition (1), and condition (1) is only sufficient, not necessary, for deletion-tolerance.
• The Gaussian zero process on the hyperbolic plane is both insertion-tolerant and deletion-tolerant, as shown via convergence of conditional laws and Rouché’s theorem.
• The theory extends to general locally compact second-countable Hausdorff groups with invariant measures, allowing application to non-Euclidean spaces like the hyperbolic plane.