Spectra of Poisson functionals and applications in continuum percolation

TL;DR

This paper introduces a spectral point process for Poisson functionals, enabling analysis of noise sensitivity and stability in continuum percolation models, with applications to Boolean and Voronoi percolation.

Contribution

It defines a new spectral point process for Poisson functionals and connects it to existing spectral concepts, advancing the understanding of noise properties in continuum percolation.

Findings

Proves sharp noise instability for crossing events in Poisson Boolean percolation.

Establishes sharp noise sensitivity for crossing events in Poisson Voronoi percolation.

Demonstrates quasi-multiplicativity of 4-arm probabilities in critical Poisson Boolean percolation.

Abstract

Let $\eta$ be a Poisson random measure (defined on some Polish space), and let $F(\eta)$ be a square-integrable functional of $\eta$. In this paper we define and study a new notion of {\it spectral point process} associated with $F(\eta)$, and use such an object to study sharp noise instability and sensitivity properties of planar critical continuum percolation models under spatial birth-death (Ornstein-Uhlenbeck) dynamics -- the notion of sharp noise instability being a natural strengthening of the absence of noise stability. The concept of spectral point process is defined by exploiting the Wiener-It\^o chaos expansion of $F$, and represents a natural continuum counterpart to the notion of {\it spectral sample}, as introduced in Garban, Pete and Schramm (2010), in the context of discrete percolation models. In the particular case where $\eta$ is a marked Poisson measure, we use Hoeffding-ANOVA decompositions to establish an explicit connection with the notion of {\it annealed spectral sample}, introduced in Vanneuville (2021) in the context of Poisson-Voronoi percolation. We also relate spectral processes with an appropriate notion of {\it pivotal processes}. As applications, we show sharp noise instability of crossing events in the critical Poisson Boolean model with unit-radius balls and, using an observation of Vanneuville, we obtain sharp noise sensitivity (as well as sharp noise instability) for crossing events in the Poisson Voronoi percolation model. As an important ingredient, we prove quasi-multiplicativity of the $4$-arm probabilities in the critical Poisson Boolean percolation model.