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Long time behaviour of one facilitated kinetically constrained models: results and open problems

Fabio Martinelli, Assaf Shapira, Cristina Toninelli

TL;DR

This work analyzes the long-time behavior of the one-facilitated kinetically constrained model FA-1f and closely related processes on ${\\mathbb Z}$, focusing on stationary measures and convergence to equilibrium in regimes with small infection density $q$. It develops a bootstrap-percolation framework to classify ergodic components and derives explicit decompositions of invariant measures in 1D (e.g., $\\mu=\\lambda\\pi+(1-\\lambda)\\underline{\\mathbf{1}}$ for FA-1f) and richer East-type mixtures, with higher-dimensional extensions conjectured through $\\mu=\\int_{\\mathcal E} d\\mu_*(\\beta)\\pi_{\\beta}$. The paper also establishes exponential relaxation results for the delta-West process and for BABP via duality with the Double Flip Process (DFP), including an extension to higher dimensions; it further analyzes FA-1f starting from finitely many vacancies, proving linear-in-time growth of the infection front. Overall, it advances understanding of stationary structure and non-stationary relaxation in constrained dynamics, highlighting techniques that combine bootstrap-percolation, detailed balance in 1D, and duality-based ergodicity proofs to tackle non-attractive stochastic dynamics.

Abstract

Kinetically constrained models (KCMs) are interacting particle systems introduced in the '80s by physicists to have accessible stochastic models with glassy-type dynamics. The key mechanism behind the complex evolution of these otherwise simple models is the so-called dynamical facilitation, a feature embedded into the models via appropriate kinetic constraints. KCMs are reversible with respect to a Bernoulli product measure, and the analysis of their stationary evolution has witnessed significant progress in the last decade. Unfortunately, in the interesting regime when the equilibrium density of the facilitating vertices is small, many fundamental questions concerning the non-stationary evolution of even the simplest models remain unsolved. In this paper, we discuss some of these questions, along with partial new results and conjectures, for the one facilitated model and its variants, as well as for the biased annihilating branching process.

Long time behaviour of one facilitated kinetically constrained models: results and open problems

TL;DR

This work analyzes the long-time behavior of the one-facilitated kinetically constrained model FA-1f and closely related processes on , focusing on stationary measures and convergence to equilibrium in regimes with small infection density . It develops a bootstrap-percolation framework to classify ergodic components and derives explicit decompositions of invariant measures in 1D (e.g., for FA-1f) and richer East-type mixtures, with higher-dimensional extensions conjectured through . The paper also establishes exponential relaxation results for the delta-West process and for BABP via duality with the Double Flip Process (DFP), including an extension to higher dimensions; it further analyzes FA-1f starting from finitely many vacancies, proving linear-in-time growth of the infection front. Overall, it advances understanding of stationary structure and non-stationary relaxation in constrained dynamics, highlighting techniques that combine bootstrap-percolation, detailed balance in 1D, and duality-based ergodicity proofs to tackle non-attractive stochastic dynamics.

Abstract

Kinetically constrained models (KCMs) are interacting particle systems introduced in the '80s by physicists to have accessible stochastic models with glassy-type dynamics. The key mechanism behind the complex evolution of these otherwise simple models is the so-called dynamical facilitation, a feature embedded into the models via appropriate kinetic constraints. KCMs are reversible with respect to a Bernoulli product measure, and the analysis of their stationary evolution has witnessed significant progress in the last decade. Unfortunately, in the interesting regime when the equilibrium density of the facilitating vertices is small, many fundamental questions concerning the non-stationary evolution of even the simplest models remain unsolved. In this paper, we discuss some of these questions, along with partial new results and conjectures, for the one facilitated model and its variants, as well as for the biased annihilating branching process.
Paper Structure (22 sections, 26 theorems, 87 equations)

This paper contains 22 sections, 26 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. State of the art and some conjectures
  3. Notation
  4. The Markov process
  5. One dimensional examples
  6. The one facilitated process (FA-1f )
  7. The biased annihilating branching process (BABP)
  8. The East polluted FA-1f
  9. The weakly unoriented East process (or $\delta$-West process)
  10. A general result on the stationary measures
  11. Proof of Theorem \ref{['thm:1d_detailed_balance']}
  12. Proof of Lemma \ref{['lem:internally_spanned']}
  13. Proof of Theorem \ref{['thm:emptiable_stationary_measure']}
  14. Proof of Lemma \ref{['lem:externally_spanned']}
  15. Proof of Theorem \ref{['thm:east_stationary_measures']}
  16. ...and 7 more sections

Key Result

Theorem 2.2

Two configurations $\eta$ and $\eta'$ belong to the same ergodic component (i.e., for any finite $\Lambda\subset{\mathbb Z} ^{d}$ there is a sequence of legal flips bringing $\eta$ to a configuration which agrees with $\eta'$ on $\Lambda$) if and only if $\operatorname{BP}^{}(\eta)=\operatorname{BP}

Theorems & Definitions (65)

  • Conjecture 1
  • Conjecture 2
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 55 more