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Mixture representation of stationary measures for general KCM via bootstrap percolation components

Prove that for any kinetically constrained model (KCM), every stationary measure μ can be represented as μ = ∫_{β∈𝔈} π_β dμ_*(β), where 𝔈 is the set of configurations stable under the associated bootstrap percolation, μ_* is a probability measure on 𝔈, and π_β is the Bernoulli(q) product measure conditioned on the bootstrap percolation limit being β.

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Background

Stationary measures in KCMs are intimately connected to the ergodic components determined by bootstrap percolation limits. The paper proves detailed balance for stationary measures (in 1D without translation invariance) and identifies stationary measures explicitly for several models (e.g., East, BABP, δ-West) via internal/external spanning arguments.

Motivated by these identifications, the conjecture posits a general mixture representation: every stationary measure is a mixture of product measures conditioned on the bootstrap-percolation-stable configurations. Establishing this would unify the characterization of stationary states across KCMs and bridge dynamics with bootstrap percolation structure.

References

In view of the discussion above and of Theorem \ref{thm:d-East}, we propose the following conjecture. Let $\mu$ be a stationary measure of a KCM. Then there exists a probability measure $\mu_{}$ on $\mathcal{E}$ such that $\mu=\int_{\mathcal{E}} d\mu_{}(\beta)\pi_{\beta}$, where $\pi_{\beta}$ is given by the measure $\pi$ conditioned on the event ${\bp{\eta}{}=\beta}$.

— Long time behaviour of one facilitated kinetically constrained models: results and open problems (2510.20461 - Martinelli et al., 23 Oct 2025) in Section 2 (A general result on the stationary measures), Conjecture 1bis