Asymptotic Behaviors of The Size of The Largest Cluster in One Dimensional Percolation

TL;DR

This paper investigates the asymptotic properties of the largest cluster size in one-dimensional percolation, revealing a critical phenomenon and analyzing its limit distribution in finite iid Bernoulli sequences.

Contribution

It introduces a detailed analysis of the asymptotic behavior and limit distribution of the largest cluster length in 1D percolation, highlighting a critical phenomenon.

Findings

Identification of a critical phenomenon in cluster length

Derivation of the limit distribution for the largest cluster

Insights into phase transition behavior in 1D percolation

Abstract

This paper focuses on the asymptotic behaviors of the length of the largest 1-cluster in a finite iid Bernoulli sequence. We first reveal a critical phenomenon on the length and then study its limit distribution.