Uniqueness of the infinite cluster for monotone percolation models without insertion tolerance

Abstract: We consider a broad class of dependent site-percolation models on $\mathbb{Z}^d$ obtained by applying a monotone automaton to a random initial particle configuration drawn from a stochastically increasing family of measures. We prove that whenever the underlying particle configuration is sampled from an insertion-tolerant measure and the avalanches generated by the dynamics produce connected sets, the supercritical phase almost surely contains a unique infinite cluster. Our result applies to several well-studied interacting particle systems, including the Abelian sandpile, activated random walk, and bootstrap percolation. In these models, the induced percolation measure typically does not satisfy standard conditions such as finite energy or insertion tolerance, so the classical Burton-Keane argument does not apply. As an application, we answer a question of Fey, Meester, and Redig concerning percolation of the toppled vertices in the Abelian sandpile with i.i.d. initial configuration.

• The paper establishes unique infinite cluster existence in the supercritical phase using auxiliary insertion-tolerant coupling, mass-transport, and multi-valued map arguments.
• It tackles dependent percolation models without classical insertion tolerance, resolving open questions for systems like the Abelian sandpile and bootstrap percolation.
• The results offer robust predictions for connectivity in systems with absorbing-state transitions and self-organized criticality, guiding future studies on complex spatial processes.

Uniqueness of the Infinite Cluster for Monotone Percolation Without Insertion Tolerance

Overview

This paper addresses the problem of uniqueness of the infinite cluster in monotone, dependent site-percolation models on $\mathbb{Z}^d$ generated via monotone automata acting on random initial configurations. Critically, these induced percolation measures can lack the insertion-tolerance property, which precludes the use of classical uniqueness arguments. The work introduces a robust comparison framework employing an auxiliary, insertion-tolerant model, together with the mass-transport principle and multi-valued map arguments, to establish that, under natural conditions, the supercritical phase almost surely contains a unique infinite cluster. This resolves longstanding questions for models such as the Abelian sandpile, activated random walks, and bootstrap percolation.

Theoretical Context and Motivation

In classical Bernoulli percolation, uniqueness of the infinite cluster above criticality is well-understood and follows from the Aizenman-Kesten-Newman and Burton-Keane arguments, which exploit independence and the so-called finite-energy or insertion-tolerance properties [AKN87; BK89]. For dependent systems, these properties are typically unavailable or fail: notably, in sandpile models, the addition of a single particle can create an uncontrolled "avalanche" affecting infinitely many sites.

The work of Fey, Meester, and Redig [FMR09] on the infinite sandpile model posed the open question of whether, for product initial measures, the percolation of toppled vertices has a unique infinite cluster. This paper answers that question affirmatively and, more broadly, develops a general strategy for a wide class of monotone, non-insertion-tolerant percolation models whose physical realizations include many paradigmatic systems with absorbing-state transitions and self-organized criticality.

Model Framework

Consider a one-parameter family of random initial particle configurations on $\mathbb{Z}^d$, with measures $\mathbb{P}_p$ that are stochastically increasing and satisfy translation-invariance, monotonicity, ergodicity, and, critically, insertion-tolerance only for the intermediate coupled measures. The percolation configuration $\omega^{p}$ is obtained by processing an initial configuration $\xi$ via a monotone automaton $T$ that outputs a site-configuration in $\{0,1\}^{\mathbb{Z}^d}$.

The class encapsulates the Abelian sandpile model, activated random walks, and monotone cellular automata (e.g., bootstrap percolation), with explicit technical conditions (D1-D4) guaranteeing translation invariance, monotonicity, an occupation threshold, and robust connectedness under site modifications. The paper demonstrates these conditions for the sandpile and related models via structural properties of topplings and the Abelian property.

Main Results

The principal theorem states:

For every $p > p_c$, there is almost surely a unique infinite cluster in $\omega^{p}$, provided the underlying initial configuration law is (intermediately) insertion-tolerant and avalanches produce connected sets.

Technical Innovations

The proof proceeds in three main steps:

1. Auxiliary Insertion-Tolerant Coupling: For parameters $p_1 < p_2 < p_3$ above criticality, construct an interpolating configuration $\mathbb{Z}^d$0 that is insertion-tolerant, ensuring, via the Burton-Keane approach, the uniqueness of its infinite cluster.
2. Comparison and Cluster Overlap Analysis: Use the mass-transport principle to analyze how hypothetical infinite clusters in $\mathbb{Z}^d$1 that do not contain the unique infinite cluster of $\mathbb{Z}^d$2 must attain their minimal distance infinitely often, leading to a contradiction.
3. Multi-Valued Map Argument (or Independence): For general dependent models, devise a careful multi-valued map to open paths and merge clusters, leveraging insertion-tolerance and monotonicity, to preclude the existence of multiple infinite clusters. For product measures, this step can be replaced by a more straightforward independence-based argument.

These techniques collectively circumvent the lack of finite energy/insertion-tolerance in the induced measures and provide a template for handling long-range dependencies.

Implications

Practical: The results provide rigorous guarantees for the uniqueness of percolating structures in a variety of models, underpinning predictions for the large-scale connectivity in systems exhibiting absorbing-state transitions and self-organized criticality, such as sandpile models and activated random walks. This strengthens the theoretical foundation for understanding the critical phase and emergent geometric properties in these systems.

Theoretical: The work generalizes classical percolation uniqueness theory to a significant class of dependent, non-insertion-tolerant processes, showing that monotonicity and the existence of well-behaved auxiliary couplings suffice to ensure uniqueness. The techniques also offer a schematic for future work in higher complexity or other graph settings (e.g., vertex-transitive amenable graphs), as discussed in the remarks.

Notably, the methodology points to a blueprint for further analysis in models where insertion-tolerance is outright unavailable, and opens the door to characterizing uniqueness and phase structure in a wider set of correlated spatial processes.

Speculation on Future Developments

Potential avenues for future research include:

• Extension to further classes of graphs beyond vertex-transitive amenable graphs, including non-unimodular or non-amenable geometries.
• Investigation of related dependent models (e.g., in continuum settings or with long-range interactions) with only partial monotonicity.
• Analysis of dynamical or evolving percolation processes without insertion tolerance, leveraging the comparison/coupling framework introduced here.
• Study of uniqueness and critical exponents for percolation of non-local or topologically complex objects generated by more general automata.

Conclusion

The paper establishes that monotone automaton-generated percolation models on $\mathbb{Z}^d$3—even in the absence of insertion-tolerance—exhibit a unique infinite cluster above criticality, under mild structural and stochastic assumptions. The coupling strategy, mass-transport, and multi-valued map argument collectively represent a robust method for dependent percolation, resolving longstanding open questions for significant interacting particle systems and suggesting new directions for the study of phase transitions in complex stochastic spatial models.

Reference:

"Uniqueness of the infinite cluster for monotone percolation models without insertion tolerance" (2603.29420)