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Electrocyclic reactions, cycloadditions, and sigmatropic rearrangements are concerted pericyclic reactions that proceed via a cyclic transition state. These reactions are stereospecific and regioselective. The stereochemistry of the products depends on the symmetry characteristics of the interacting orbitals and the reaction conditions. Accordingly, pericyclic reactions are classified as either symmetry-allowed or symmetry-forbidden. Woodward and Hoffmann presented the selection criteria for...
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Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.
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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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Multimachine stability analysis is crucial for understanding the dynamics and stability of power systems with multiple synchronous machines. The objective is to solve the swing equations for a network of M machines connected to an N-bus power system.
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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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Updated: Jun 13, 2025

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform
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Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules.

Hugo Duminil-Copin1,2, Ivailo Hartarsky3

  • 1Université de Genève, Section de Mathématiques, 2-4 rue du Lièvre, 1211 Geneva, Switzerland.

Probability Theory and Related Fields
|September 16, 2024
PubMed
Summary

This study demonstrates that certain two-dimensional bootstrap percolation models exhibit a sharp metastability transition. This finding applies to models with isotropic threshold rules and convex symmetric neighborhoods, generalizing previous results.

Keywords:
Bootstrap percolationMetastabilitySharp threshold

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Area of Science:

  • Statistical Physics
  • Probability Theory
  • Mathematical Physics

Background:

  • Bootstrap percolation models are fundamental in studying phase transitions.
  • Previous research focused on specific rules, limiting generalizability.
  • Understanding metastability is crucial for complex systems.

Purpose of the Study:

  • To establish a sharp metastability transition for a broader class of 2D bootstrap percolation models.
  • To generalize findings beyond specific, previously studied rules.
  • To provide a contemporary perspective on bootstrap percolation theory.

Main Methods:

  • Analysis of two-dimensional critical bootstrap percolation models.
  • Focus on isotropic threshold rules with convex symmetric neighborhoods.
  • Mathematical proofs establishing the transition.

Main Results:

  • A class of bootstrap percolation models exhibits a sharp metastability transition.
  • This includes all isotropic threshold rules with convex symmetric neighborhoods.
  • The findings extend prior results for specific percolation rules.

Conclusions:

  • The identified class of models demonstrates universal metastability behavior.
  • This work unifies and extends previous specific findings.
  • The study offers a modern framework for bootstrap percolation universality.