The longest edge in discrete and continuous long-range percolation

TL;DR

This paper analyzes the asymptotic distribution of the longest edge in long-range percolation models, revealing different extreme value behaviors depending on connection probabilities, with implications for understanding large-scale connectivity.

Contribution

It provides a comprehensive extreme value analysis of the longest edge in both continuous and discrete long-range percolation models, including new regimes and singularities.

Findings

Longest edge length follows different extreme value distributions based on connection probability.

Established a formal construction using marked Poisson processes and coupling arguments.

Identified parameter regimes with unique behaviors and unexpected singularities.

Abstract

We consider the random connection model in which an edge between two Poisson points at distance $r$ is present with probability $g(r)$. We conduct an extreme value analysis on this model, namely by investigating the longest edge with at least one endpoint within some finite observation window, as the volume of this window tends to infinity. We show that the length of the latter, after normalizing by some appropriate centering and scaling sequences, asymptotically behaves like one of each of the three extreme value distributions, depending on choices of the probability $g(r)$. We prove our results by giving a formal construction of the model by means of a marked Poisson point process and a Poisson coupling argument adapted to this construction. In addition, we study a discrete variant of the model. We obtain parameter regimes with varying behavior in our findings and an unexpected singularity.