Characterizing spatial point processes by percolation transitions

TL;DR

This paper investigates the nature of percolation transitions in spatial point processes across various dimensions, examining ensemble equivalence and universal properties to classify natural point patterns.

Contribution

It introduces a canonical ensemble framework for analyzing continuum percolation, contrasting it with the traditional grand-canonical approach, and explores universality in percolation transitions.

Findings

Canonical ensemble describes percolation transitions effectively.

Universal properties of percolation are consistent across different point processes.

Provides a new method to classify natural spatial point patterns.

Abstract

A set of discrete individual points located in an embedding continuum space can be seen as percolating or non-percolating, depending on the radius of the discs/spheres associated with each of them. This problem is relevant in theoretical ecology to analyze, e.g., the spatial percolation of a tree species in a tropical forest or a savanna. Here, we revisit the problem of aggregating random points in continuum systems (from $2$ to $6-$dimensional Euclidean spaces) to analyze the nature of the corresponding percolation transition in spatial point processes. This problem finds a natural description in terms of the canonical ensemble but not in the usual grand-canonical one, customarily employed to describe percolation transitions. This leads us to analyze the question of ensemble equivalence and study whether the resulting canonical continuum percolation transition shares its universal properties with standard percolation transitions, analyzing diverse homogeneous and heterogeneous spatial point processes. We, therefore, provide a powerful tool to characterize and classify a vast class of natural point patterns, revealing their fundamental properties based on percolation phase transitions.