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Fredrickson--Andersen model in two dimensions

Ivailo Hartarsky, Fabio Martinelli, Cristina Toninelli

Abstract

The present expository article overviews recent mathematical advances on the Fredrickson--Andersen kinetically constrained spin model in two dimensions. It was introduced in physics as a toy model for recovering the glassy phenomenology in supercooled liquids close to the glass transition via dynamic constraints as opposed to static interactions.

Fredrickson--Andersen model in two dimensions

Abstract

The present expository article overviews recent mathematical advances on the Fredrickson--Andersen kinetically constrained spin model in two dimensions. It was introduced in physics as a toy model for recovering the glassy phenomenology in supercooled liquids close to the glass transition via dynamic constraints as opposed to static interactions.
Paper Structure (14 sections, 4 theorems, 20 equations, 1 figure)

This paper contains 14 sections, 4 theorems, 20 equations, 1 figure.

Table of Contents

  1. Models
  2. Bootstrap percolation
  3. The Fredrickson--Andersen model
  4. FA-1f
  5. An auxiliary model: CBSEP
  6. 2-neighbour bootstrap percolation
  7. Coarse threshold
  8. Sharp threshold
  9. Speed of convergence
  10. FA-2f
  11. Background
  12. Result
  13. Behind Theorem \ref{['th:FA2f']}: high-level ideas
  14. Open problems

Key Result

Proposition 2.1

If $G={\mathbb Z} ^2$, then $\IfNoValueTF {CBSEP} {{T_{\rm rel}} }{{T_{\rm rel}^{\mathrm{{CBSEP}}}}}\leqslant O(\log(1/p)/p)$.

Figures (1)

  • Figure 1: Recursive structure of droplets used in 2-neighbour bootstrap percolation Holroyd03 and FA-2f Hartarsky20FA respectively. Arrows indicate regions with no two consecutive healthy lines.

Theorems & Definitions (6)

  • Remark 1.1
  • Proposition 2.1
  • Lemma 3.1: Aizenman--Lebowitz Aizenman88
  • Theorem 3.2: Second term
  • Theorem 4.1
  • Remark 4.2