Threshold-Controlled Geometric Reorganization in 2D Bootstrap Percolation

TL;DR

This paper investigates how increasing the activation threshold in 2D bootstrap percolation causes a significant geometric reorganization of the absorbing state, shifting from collective propagation to boundary-driven stabilization.

Contribution

It reveals a threshold-controlled geometric crossover in bootstrap percolation, highlighting boundary organization as a key structural feature at high thresholds.

Findings

At low thresholds, observables are confined to a single low-p window.

High thresholds cause observables to split into distinct geometric response scales.

Finite-size effects shift from active density fluctuations to boundary observables.

Abstract

Two-dimensional bootstrap percolation is usually characterized by bulk observables, but whether increasing the activation threshold qualitatively reorganizes the geometry of the absorbing state has remained unclear. Here we show that the response undergoes a threshold-controlled geometric crossover. At low thresholds, the extrema of bulk and boundary-sensitive observables remain confined to a single collective low-$p$ window. At high thresholds, they split into distinct branches, revealing multiple geometric response scales. Over the accessible system sizes, the dominant finite-size signatures shift from fluctuations of the final active density to non-singleton boundary observables, while the fluctuation peak itself decreases. Time-resolved mechanism traces show that this crossover is accompanied by a progression from extended collective propagation to frontier exhaustion and, at the highest threshold, to quasi-one-step stabilization. Our results identify boundary organization as the dominant structural signature of high-threshold bootstrap percolation and show that conventional bulk observables alone do not capture the full reorganization of the absorbing state.