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Locality Approach to the Bootstrap Percolation Paradox.

Ivailo Hartarsky1, Augusto Teixeira2

  • 1Technische Universität Wien, Institut für Stochastik und Wirtschaftsmathematik, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria.

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Summary

This study resolves discrepancies in bootstrap percolation models by linking mathematical and local counterparts. New methods achieve precise agreement between simulations and theory, enabling novel predictions.

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Area of Science:

  • Statistical Physics
  • Probability Theory
  • Mathematical Modeling

Background:

  • The bootstrap percolation model is a fundamental concept in statistical physics and network theory.
  • Previous theoretical predictions and Monte Carlo simulations showed significant discrepancies, even in asymptotic behavior.
  • Understanding these differences is crucial for accurately modeling emergent phenomena.

Purpose of the Study:

  • To reconcile the long-standing disagreements between theoretical predictions and simulation results in the bootstrap percolation model.
  • To introduce a novel mathematical framework that connects the global bootstrap percolation model with its local counterpart.
  • To establish a foundation for generating new, accurate predictions for the model's behavior.

Main Methods:

  • Leveraging recent mathematical advancements to link the bootstrap percolation model with its local counterpart.
  • Developing a new theoretical framework to analyze the model's behavior.
  • Comparing theoretical predictions with Monte Carlo simulations up to the third-order expansion.

Main Results:

  • The new framework successfully resolves historic discrepancies between theoretical results and Monte Carlo simulations.
  • Excellent agreement between numerical simulations and theoretical predictions is achieved, particularly as the infection probability approaches zero.
  • The agreement extends to the third-order expansion, a significant improvement over previous findings.

Conclusions:

  • The developed mathematical approach provides a unified perspective on bootstrap percolation, bridging theory and simulation.
  • This work sets a new standard for accuracy in modeling bootstrap percolation, with implications for network science.
  • The methodology enables the generation of novel, reliable predictions for the bootstrap percolation model.