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

This paper establishes the almost sure uniqueness of the infinite cluster in the supercritical phase for a broad class of dependent site-percolation models on $\mathbb{Z}^d$ generated by monotone automata acting on insertion-tolerant initial configurations, thereby extending uniqueness results to systems like the Abelian sandpile and bootstrap percolation where classical arguments fail due to the lack of insertion tolerance.

The Gist

The Big Picture: A Game of Dominoes and Snowballs

Imagine a giant, infinite grid of streetlights (a lattice). Some lights are on, some are off. In the world of percolation theory, mathematicians ask a simple question: If enough lights are turned on, will they connect to form one giant, infinite highway of light that stretches forever?

Usually, the answer is "Yes, and there is only one such highway." But this paper tackles a much trickier version of the game where the lights don't just turn on randomly. Instead, they are part of a complex system where turning on one light can trigger a chain reaction, like a domino effect or an avalanche.

The authors, Christoforos Panagiotis and Alexandre Stauffer, prove that even in these chaotic, chain-reaction systems, there is still only one giant infinite highway.

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The Problem: When the Rules Break

In standard games (like flipping a coin for every light), mathematicians have a famous tool called the Burton-Keane argument to prove there's only one infinite highway. It relies on a rule called "insertion tolerance."

The Analogy:
Imagine you are building a wall. "Insertion tolerance" means that if you add one extra brick, the wall doesn't collapse or change its entire shape drastically. You can tweak the system locally without breaking the global rules.

The Issue:
The models in this paper (like the Abelian Sandpile, Activated Random Walk, and Bootstrap Percolation) are like a house of cards. If you add a single grain of sand to a pile, it might trigger a massive avalanche that topples the whole structure. Because adding one particle can cause an infinite chain reaction, the standard "Burton-Keane" tool breaks. It's like trying to use a ruler to measure a volcano; the tool isn't designed for that kind of explosion.

For years, mathematicians wondered: If the rules are this chaotic, can we still be sure there's only one infinite cluster, or could there be two or three separate infinite highways running parallel to each other?

The Solution: The "Bridge" Strategy

The authors developed a clever three-step strategy to solve this without needing the standard rules. Think of it as building a bridge between two islands to prove they are actually connected.

Step 1: Build a "Safe" Bridge

They imagine three different versions of the game:

1. Low Density ($p_1$): Not many lights are on.
2. Medium Density ($p_2$): More lights are on.
3. High Density ($p_3$): Lots of lights are on (this is the real system we care about).

They create a hybrid system (a bridge) between the Low and Medium versions. They tweak the rules just enough so that this hybrid system does obey the "safe" rules (insertion tolerance).

• The Result: Because this hybrid system is "safe," we know for a fact it has exactly one infinite highway. Let's call this the "Golden Highway."

Step 2: The Mass-Transport Detective Work

Now, they look at the real, chaotic High-Density system ($p_3$). They ask: Could there be a second, separate infinite highway in the real system that doesn't touch the Golden Highway?

They use a mathematical tool called the Mass-Transport Principle.

• The Analogy: Imagine people living on the infinite grid. If there were two separate infinite highways, people living on the "stranger" highway would have to constantly look across the gap to the "Golden Highway."
• The authors prove that if a second highway exists, the distance between the two highways would have to be "measured" an infinite number of times by the people living on them.
• The Catch: In a stable, infinite grid, you can't have a situation where everyone is constantly measuring the same distance over and over again without running out of "energy" or violating the laws of probability. It's like a town where everyone is constantly shouting the same number to a neighbor across a canyon; eventually, the math says this is impossible.

Step 3: The "Multi-Valued Map" Trap

To seal the deal, they use a logic trick called a Multi-Valued Map.

• The Analogy: Imagine you have a map of the city. You find a spot where the two highways are closest. You then try to "rewind" the game by removing a few particles to see if the highways merge.
• They show that if you try to force the two highways to stay separate, you end up with a logical contradiction. It's like trying to prove that two rivers are separate, but every time you trace their paths, you find they must have merged upstream.
• Because the "avalanches" in these systems always create connected sets (the dominoes fall in a continuous line), the two highways cannot stay apart. They are forced to merge into one.

Why Does This Matter?

This paper solves a specific mystery about the Abelian Sandpile Model.

• The Sandpile: Imagine a pile of sand on a grid. When a pile gets too high, it topples, sending grains to neighbors, which might make them topple.
• The Question: If you start with a random pile of sand, and let it settle, will the spots that "toppled" (moved) form one giant connected web, or many separate webs?
• The Answer: Fey, Meester, and Redig asked this question years ago. This paper says: It's always one giant web.

Summary in One Sentence

Even in chaotic systems where a single change can trigger a massive, unpredictable avalanche, if the chaos spreads in a connected way, nature still ensures that there is only one giant, infinite structure, not many competing ones.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in percolation theory: Does the supercritical phase of a dependent site-percolation model contain a unique infinite cluster?

• Context: In standard Bernoulli percolation (independent sites), the uniqueness of the infinite cluster is a classical result (Aizenman-Kesten-Newman, Burton-Keane). The Burton-Keane argument relies heavily on the finite-energy or insertion tolerance property, which asserts that the state of a vertex can be changed with finite cost (positive probability) without affecting the rest of the system.
• The Gap: Many important interacting particle systems (e.g., Abelian sandpiles, activated random walks, bootstrap percolation) generate dependent percolation configurations that lack insertion tolerance. In these models, adding a single particle can trigger an "infinite avalanche," fundamentally altering the global configuration. Consequently, standard uniqueness proofs fail.
• Specific Motivation: The authors aim to resolve a specific open question posed by Fey, Meester, and Redig regarding the Abelian sandpile: Is the infinite cluster of toppled vertices unique when the initial configuration is sampled from a product measure (i.i.d.)?

2. Model Definitions and Framework

The authors define a broad class of models on the hypercubic lattice $\mathbb{Z}^d$ (and more generally, vertex-transitive amenable graphs) consisting of two components:

1. Underlying Randomness ($P_p$): A one-parameter family of stochastically increasing measures on particle configurations $\xi \in [0, \infty)^{\mathbb{Z}^d}$. These measures satisfy:

   • Translation Invariance (R1) and Ergodicity (R4).
   • Monotonicity (R2): A coupling exists such that $X \le Y \le Z$ for $p_1 < p_2 < p_3$.
   • Insertion Tolerance (R3): While the final percolation state may not be insertion tolerant, the underlying particle measures allow for local increases (adding particles) with positive probability, provided the increase is bounded by a threshold $t$.

2. Dynamics ($T$): A map $T: [0, \infty)^{\mathbb{Z}^d} \to \{0, 1\}^{\mathbb{Z}^d}$ that transforms the particle configuration into a percolation configuration $\omega$. The map must satisfy:

   • Translation Invariance (D1) and Monotonicity (D2).
   • Occupation Threshold (D3): If a site has $\ge t$ particles, it is "open" ($\omega_x=1$).
   • Connectivity (D4): Increasing the particle count at a site $x$ may change the cluster containing $x$, but it cannot create or destroy infinite clusters elsewhere. Crucially, this implies that the set of newly open vertices forms a connected set (a property satisfied by Abelian sandpiles and activated random walks).

3. Methodology

The proof of uniqueness (Theorem 1.1) is structured in three distinct steps, utilizing a combination of coupling, mass transport, and multi-valued map arguments.

Step 1: Construction of an Insertion-Tolerant Interpolant

Since the original model $\omega_p$ lacks insertion tolerance, the authors construct an auxiliary configuration $\omega_{p_1, p_2}$ for parameters $p_1 < p_2$.

• Definition: $\omega_{p_1, p_2}(x) = 1$ if $\omega_{p_1}(x) = 1$ OR if the underlying particle count $Y_x \ge t$.
• Property: This auxiliary model is insertion-tolerant because the underlying measure $Y$ allows local increases.
• Result: By the classical Burton-Keane argument, $\omega_{p_1, p_2}$ possesses a unique infinite cluster (denoted $C^\infty_{p_1, p_2}$) almost surely.

Step 2: Mass-Transport Principle and Distance Analysis

The authors relate the auxiliary cluster to the original supercritical cluster $\omega_{p_3}$ (where $p_3 > p_2$).

• Hypothesis: Assume there exists an infinite cluster $C$ in $\omega_{p_3}$ that does not contain $C^\infty_{p_1, p_2}$.
• Argument: Using the Mass-Transport Principle, they analyze the distance between $C$ and $C^\infty_{p_1, p_2}$.
• Contradiction: If the distance were attained finitely many times, it would violate the mass-transport balance (specifically, the sum of flows would be finite on one side and infinite on the other).
• Conclusion: The distance between any disjoint infinite cluster in $\omega_{p_3}$ and the unique cluster in $\omega_{p_1, p_2}$ must be attained infinitely many times.

Step 3: Multi-Valued Map Argument (The Core Innovation)

To prove that no such disjoint cluster $C$ can exist, the authors employ a multi-valued map technique to merge clusters.

• Setup: Assume with positive probability that an infinite cluster $C^\infty_{p_3}$ exists at a fixed distance $D$ from $C^\infty_{p_1, p_2}$, with the distance attained $K$ times within a large box.
• The Map: They define a map that modifies the underlying particle configuration $\xi_{p_3}$ along geodesics connecting the two clusters. Specifically, they force particles to exceed the threshold $t$ along these paths, effectively "filling the gap" and merging the clusters.
• Counting:
  • Forward: The insertion tolerance of the underlying measure ensures that the probability of the modified configuration is bounded below by $\epsilon^D$.
  • Backward: They bound the number of pre-images (how many original configurations map to the same modified one). Due to the connectivity constraint (D4), the number of pre-images is polynomial in the box size, while the number of "merging opportunities" ($K$) can be chosen arbitrarily large.
• Contradiction: By choosing $K$ sufficiently large, the probability of the merged event exceeds the probability of the original event, leading to a contradiction. Thus, the assumption that disjoint infinite clusters exist is false.

4. Key Results

• Theorem 1.1: For any model satisfying the defined properties (R1-R4, D1-D4), if the underlying measure is insertion-tolerant (or i.i.d.) and avalanches produce connected sets, then for every $p > p_c$, the number of infinite clusters $N = 1$ almost surely.
• Generalization: The result holds not only for $\mathbb{Z}^d$ but for any vertex-transitive amenable graph (Remark 3.1).
• Specific Application: The theorem confirms that for the Abelian sandpile with i.i.d. initial configurations, the set of toppled vertices forms a unique infinite cluster in the supercritical phase, answering the question of Fey, Meester, and Redig.

5. Significance and Contributions

• Overcoming the Insertion Tolerance Barrier: This is the first general proof of infinite cluster uniqueness for a broad class of dependent percolation models that explicitly fail the insertion tolerance condition.
• Unified Framework: The paper provides a rigorous framework (Properties R1-R4, D1-D4) that encompasses diverse systems like the Abelian sandpile, activated random walks, and bootstrap percolation, treating them under a single theoretical umbrella.
• Methodological Advancement: The combination of the Burton-Keane argument (applied to an auxiliary interpolated model) with a multi-valued map argument (to handle the lack of independence in the original model) represents a significant technical innovation in percolation theory.
• Resolution of Open Problems: It settles long-standing questions regarding the topology of clusters in self-organized criticality models, which were previously thought to be intractable using standard tools.

In summary, the paper demonstrates that even in highly dependent systems where local changes can cause global cascades, the macroscopic structure of the infinite cluster remains unique, provided the dynamics preserve connectivity and the underlying randomness is sufficiently rich.