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Locality approach to the bootstrap percolation paradox

Ivailo Hartarsky, Augusto Teixeira

TL;DR

This work reframes bootstrap percolation through a rigorous locality lens, proving that for FBP the non-locality $\log\rho - \log\rho_\ell$ is polylogarithmic in $1/p$, and thus $\rho \sim \rho_\ell$ as $p\to 0$. By combining a GPU-accelerated dynamic programming scheme with meticulous data analysis, the authors extract precise first-, second-, and third-order asymptotics, matching rigorous predictions and Monte Carlo observations for both FBP and MBP, while clarifying historical discrepancies. The methodology yields explicit constants, reveals distinct scaling corrections between FBP and MBP, and provides a practical framework to probe other BP models and higher dimensions. Overall, the locality approach reconciles theory and numerics, offering a reliable protocol to predict metastability-driven nucleation in two-dimensional BP and related systems. $\Lambda = C\log(1/p)/p$ plays a central role in the droplet construction and scaling analyses.

Abstract

We revisit the Bootstrap Percolation model, leveraging recent mathematical advances linking it with its local counterpart. This new perspective resolves, for the first time, historic discrepancies between Monte Carlo simulations and theoretical results: previously, those predictions disagreed even in the first-order asymptotics of the model. In contrast, our framework achieves excellent agreement between numerics and theory, which now match up to the third-order expansion, as the infection probability approaches zero. Our algorithm allows us to generate novel predictions for the model.

Locality approach to the bootstrap percolation paradox

TL;DR

This work reframes bootstrap percolation through a rigorous locality lens, proving that for FBP the non-locality is polylogarithmic in , and thus as . By combining a GPU-accelerated dynamic programming scheme with meticulous data analysis, the authors extract precise first-, second-, and third-order asymptotics, matching rigorous predictions and Monte Carlo observations for both FBP and MBP, while clarifying historical discrepancies. The methodology yields explicit constants, reveals distinct scaling corrections between FBP and MBP, and provides a practical framework to probe other BP models and higher dimensions. Overall, the locality approach reconciles theory and numerics, offering a reliable protocol to predict metastability-driven nucleation in two-dimensional BP and related systems. plays a central role in the droplet construction and scaling analyses.

Abstract

We revisit the Bootstrap Percolation model, leveraging recent mathematical advances linking it with its local counterpart. This new perspective resolves, for the first time, historic discrepancies between Monte Carlo simulations and theoretical results: previously, those predictions disagreed even in the first-order asymptotics of the model. In contrast, our framework achieves excellent agreement between numerics and theory, which now match up to the third-order expansion, as the infection probability approaches zero. Our algorithm allows us to generate novel predictions for the model.
Paper Structure (15 sections, 2 equations, 4 figures, 2 tables)

This paper contains 15 sections, 2 equations, 4 figures, 2 tables.

Table of Contents

  1. Models
  2. Background
  3. Rigorous locality approach
  4. What went wrong?
  5. Conclusion
  6. Acknowledgments
  7. Exact computation for local densities
  8. Monte Carlo simulation of non-local densities
  9. First order scaling
  10. First order constant
  11. Second order scaling
  12. Second order constant
  13. Third order scaling
  14. Third order constant
  15. Non-locality

Figures (4)

  • Figure 1: FBP: infection probability $0.06$, side length $93$.
  • Figure 2: MBP: infection probability $0.047$, side length $130$.
  • Figure 4: FBP: The solid line and curve are the rigorous first and second order asymptotics. The estimated third order is dashed.
  • Figure 5: MBP: The solid line depicts the second and third order terms. The dashed curve incorporates the estimated fourth order correction. The dotted box indicates the previously explored range DeGregorio06.