Mixing times of spin systems on dynamical percolation

Abstract: We study the mixing times of stochastic spin systems corresponding to nearest-neighbour Glauber dynamics on dynamical percolation, defined on $d$-dimensional torus of side-length $N$. In this model, the status of each edge (open or closed) updates independently at rate $λ>0$, according to $\mathrm{Ber}(p)$ samples. Simultaneously, the spin of each site updates at rate $1$ according to Glauber dynamics on the environment restricted to open edges. We show that for a relatively general class of nearest-neighbour systems, as long as $p<p_c(d)$, for any temperature, if $λ$ is sufficiently small, the mixing time is of order $\frac{\log N}λ$. This Markov chain is non-reversible, and the proof is obtained by developing a particular coupling that couples together local configurations whenever the environment behaves well.

• The paper demonstrates that for subcritical percolation (p < p_c), the mixing time of coupled spin systems scales as Θ((1/λ) log N).
• It introduces a novel multiscale, renormalization-based coupling method to address non-reversibility in dynamically evolving environments.
• The findings imply that slow environmental updates can eliminate metastable bottlenecks, significantly accelerating mixing even at low temperatures.

Mixing Times of Spin Systems on Dynamical Percolation: An Expert Overview

Introduction and Model Formulation

The paper "Mixing times of spin systems on dynamical percolation" (2607.02477) investigates the ergodic convergence properties—specifically, mixing times—of stochastic nearest-neighbor spin systems (including the Glauber dynamics for models such as the Ising model) evolving in a dynamically changing percolation environment. The environment is modeled as dynamical bond percolation on the $d$-dimensional discrete torus of side length $N$, with each edge resampled at rate $\lambda$ according to Bernoulli($p$). The spins at each vertex update asynchronously at rate $1$, following Glauber dynamics restricted to the current open edges.

Crucially, the process is non-reversible: the coupled evolution of spins and the underlying percolation configuration generates a Markov chain on $S^{V}\times\{0,1\}^{E}$ which does not admit a reversible invariant measure, since the transitions depend asymmetrically on the evolving environment. Analyzing mixing in such non-reversible and partially random environments is a significant technical challenge, especially as standard spectral methods often fail.

Main Theoretical Results

The primary result is a rigorous characterization of the mixing time $t_{\mathrm{mix}}$ of the joint process $(\sigma_t, \eta_t)$ for a wide class of finite-range nearest-neighbor spin systems (including all temperatures of the Ising model), provided the edge percolation parameter $p$ is subcritical ($p < p_c(d)$, where $N$0 is the critical threshold for $N$1-dimensional bond percolation).

Key statements:

• For any temperature and all $N$2, there exist constants $N$3 such that for all $N$4 and $N$5, the mixing time satisfies

$N$6

• For vanishing $N$7, the sequence of mixing times exhibits a cutoff at $N$8 with window $N$9.

Notably, this establishes that the mixing is dictated by the slowest relevant time scale: the evolution rate of the percolation environment itself ($\lambda$0), not by the intrinsic mixing of the spin subsystem. The mechanism is fully robust across system sizes and potentials, provided certain minor-technicality local mixing conditions hold.

Contradictory or notable claims:

• Unlike static Ising-Glauber dynamics, where a low-temperature phase transition leads to exponential mixing times in $\lambda$1 above a critical $\lambda$2, the coupling to a slow dynamical percolation at subcritical $\lambda$3 ensures only logarithmic dependence in $\lambda$4 for all temperatures and arbitrary interactions, as long as the percolation clusters remain finite.
• The presence of slow environment dynamics accelerates mixing for models that would otherwise display slow mixing due to spatial bottlenecks, as demonstrated by the elimination of the exponential slow phase.

Methodological Innovations and Proof Structure

The proof departs from standard spectral gap techniques, which do not apply in non-reversible, environment-coupled settings. Instead, the authors develop a multiscale, renormalization-based coupling argument:

• Time-interval partition: Time is divided into intervals of length $\lambda$5 for small $\lambda$6, exploiting the timescale separation: the percolation environment is mostly static during one such interval; conversely, the spin system is sufficiently “mixed” inside each percolation cluster over this timescale.
• Cluster-wise coupling: On subcritical percolation clusters (typically $\lambda$7 size), the local spin configuration is essentially reset, ensuring rapid local coalescence of initially different spin configurations.
• Space-time box renormalization: The space-time slab $\lambda$8 is partitioned into partially overlapping boxes, facilitating the control of disagreement propagation along oriented paths of “bad” boxes (where clusters could grow or boundary conditions are not well behaved due to atypical percolation updates).

A key technical achievement is controlling the probability of "bad" boxes by (i) exploiting the exponential tail of cluster sizes in subcritical percolation, and (ii) the tractable independence properties of percolation updates. The authors prove that the event of a disagreement percolating through the space-time slab is as unlikely as a rare oriented percolation of bad boxes, allowing the coupling to succeed globally after time $\lambda$9.

Numerical and Qualitative Implications

The result is quantitative on the asymptotic scaling, with the strong claim that the mixing time is sharply pinned to $p$0 for small $p$1 and subcritical $p$2, for all temperatures and within a broad class of interactions. This includes models which in static geometry would display torpid mixing due to spatial ergodicity breaking (e.g., low-temperature Ising).

No explicit finite-size constants are computed; however, the framework provides concrete guidance for the implementation of such coupled dynamics in simulation and sampling: to equilibrate to (near-)stationarity on a large torus, it suffices to simulate for $p$3 units of time, where $p$4 is the (chosen) percolation refresh rate. This is in stark contrast with the exponentially large timescales that may be required for fixed environments at low temperature.

Further, the cutoff argument aligns the global mixing transition sharply with the percolation environment’s mixing window, pointing towards universal scaling in such doubly-random models.

Broader Implications and Future Directions

The conclusions contribute to a growing understanding of “dynamic randomness accelerating mixing” in interacting systems—a phenomenon also seen for random walks in dynamical percolation [Peres-Stauffer-Steif 2014, Galanis-Goldberg-Mifsud 2026]. The result implies that sporadic randomization of the environment—even at low rates—can destroy metastable behavior and remove slow mixing associated with bottlenecks or phase coexistence. Such results are highly pertinent for computational approaches (e.g., perfect sampling in random environments) and for physical understanding of non-equilibrium statistical mechanics in evolving media.

The framework and methodology are expected to inform further research in:

• Mixing and cutoff for other particle systems (e.g., exclusion, voter, or contact processes) on evolving random graphs.
• Extension to supercritical regimes, where infinite clusters emerge, and the multiscale structure becomes more intricate.
• Analysis of finite but positive temperature phase transitions in the presence of weak, time-varying disorder.

Finally, this work marks a crucial step for rigorous scaling analysis of mixing in non-reversible, disordered, and dynamically coupled systems, with immediate applications in both probability theory and statistical physics.

Conclusion

This paper rigorously establishes that, for a broad class of nearest-neighbor spin systems evolving under Glauber-like dynamics on slow subcritical dynamical percolation, the global mixing time scales as $p$5 for small $p$6 and large $p$7, uniformly across temperature regimes and interaction potentials. The result elucidates how the mixing is driven by the bottleneck of the environment’s own relaxation, and develops robust coupling and renormalization arguments applicable well beyond reversible settings. These findings have important consequences for understanding ergodicity in disordered systems and for algorithmic sampling in random and evolving media (2607.02477).