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Coexistence of infinite clusters for percolation and Ising model on $\mathbb{Z}^d$

This paper establishes that for sufficiently high dimensions ( $d \geq 9$ for percolation and $d \geq 8$ for the Ising model), the complement of the infinite open cluster and the $+$ spins in the minus phase, respectively, stochastically dominate a supercritical site percolation, thereby proving the coexistence of infinite clusters and improving upon previous dimensional bounds.

Original authors: Jianping Jiang, Sike Lang

Published 2026-09-25

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CC BY 4.0

Original authors: Jianping Jiang, Sike Lang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, invisible grid stretching out in every direction, like a three-dimensional city made of points and the lines connecting them. In the world of mathematics and physics, this grid is a playground for studying how things spread. Scientists often ask a simple question: if you randomly turn on some of the connections between these points, at what point does a single, endless chain form that stretches forever? This is the heart of percolation theory, a field that helps us understand how fluids move through rock, how diseases spread through a population, or how electricity flows through a material. For decades, researchers have known that once this endless chain appears, it usually swallows up the entire space, leaving no room for anything else to grow. But a lingering question remained: is it possible for two different infinite structures to exist side-by-side in this grid, one growing through the open connections and another surviving in the gaps left behind?

This question lies at the center of a new study by Jianping Jiang and Sike Lang, who have explored the limits of this coexistence in high-dimensional spaces. Their work focuses on two famous models used to describe physical systems. The first is a model of random connections, where edges between points are either open or closed. The second is the Ising model, a classic framework for understanding magnetism, where points on the grid act like tiny magnets that can point up or down. In both cases, the researchers were looking for a specific threshold: a point just beyond the moment the first infinite structure appears, where a second, independent infinite structure might still manage to survive.

The authors proved that in grids with nine or more dimensions, such a coexistence is not only possible but guaranteed for a small range of conditions just above the critical point. Specifically, they showed that if you remove the giant, infinite cluster of open connections from the grid, the remaining points are still rich enough to form their own infinite network. In fact, this leftover network is so robust that it behaves statistically like a new, independent system where connections are formed with a high probability. This finding improves upon previous work that required even higher dimensions, pushing the boundary of certainty down to nine dimensions for the connection model.

The researchers also applied this logic to the magnetic Ising model. Here, the question is whether a region of "up" spins can form an infinite cluster even when the entire system is forced into a "down" state by its boundaries. They proved that for grids with eight or more dimensions, such an infinite cluster of up spins can indeed exist within the sea of down spins, just below the critical temperature where the system flips. Like the connection model, this result suggests that the leftover structure is strong enough to dominate a standard, independent system.

While the mathematical proofs hold firm for dimensions eight and nine and above, the authors used computer estimates to suggest that these phenomena likely occur in even lower dimensions, perhaps as low as four for the connection model and five for the magnetic model. However, they remain cautious about the lowest dimensions, noting that their specific method of proof might not work there, even though they suspect the coexistence still happens. The study relies on a clever strategy of tracking paths through the grid, allowing them to calculate the likelihood of these infinite structures forming without getting lost in the complexity of the high-dimensional space. By establishing that these two infinite worlds can share the same space, the paper deepens our understanding of how order and chaos can coexist in complex systems, revealing that the universe of mathematical grids is far more crowded and layered than previously thought.

Technical Summary: Coexistence of Infinite Clusters for Percolation and Ising Model on $\mathbb{Z}^d$

Problem Statement
The paper addresses the problem of the coexistence of infinite clusters in high-dimensional lattices $\mathbb{Z}^d$ . Specifically, it investigates whether, in the supercritical regime, the complement of the unique infinite open cluster (for bond percolation) or the set of "minority" spins (for the Ising model in the minus phase) contains an infinite connected component.

For bond percolation with parameter $p > p_c^b(d)$ , let $X_p(d)$ denote the subgraph obtained by removing all vertices belonging to the unique infinite open cluster. The central question, originally posed in [16], is whether $X_p(d)$ contains an infinite connected component. Similarly, for the Ising model at inverse temperature $\beta < \beta_c(d)$ in the minus phase (all $-$ boundary conditions), the paper investigates whether the $+$ spins form an infinite cluster.

Previous results established coexistence for very high dimensions ( $d \ge 19$ in [16], improved to $d \ge 11$ and then $d \ge 10$ in [4] and [9] respectively). For the Ising model, Aizenman, Bricmont, and Lebowitz [1] proved coexistence for sufficiently large $d$ . The goal of this work is to lower these dimensional thresholds and provide a stronger form of coexistence via stochastic domination.

Methodology
The authors employ a strategy based on stochastic domination. Instead of merely proving the existence of an infinite component, they aim to show that the remaining structure ( $X_p(d)$ or the $+$ spins) stochastically dominates a supercritical site percolation process. If a system dominates a supercritical site percolation, it necessarily contains a unique infinite cluster.

Stochastic Domination Framework:
- Percolation: The authors utilize a recent result by Martineau, Poudevigne, and Rax [29] (Theorem 3) regarding stochastic domination of random variables. They construct a mapping from the bond percolation configuration to a site percolation configuration. A key technical step involves Lemma 1, which uses Marshall Hall's extension of the marriage theorem to assign a unique incident open edge to each vertex in the infinite cluster, allowing the definition of a "blocking" mechanism.
- Ising Model: For the Ising model, the authors apply a stochastic domination result by Liggett and Steif [26] (Theorem 4). This theorem links the domination of site percolation to the decay rate of the probability that a large box is entirely filled with $-$ spins. This decay rate is related to the free energy $f_{\mathbb{Z}^d}(\beta)$ .
Upper Bounds for Critical Probabilities:
To prove that the dominated parameter exceeds the critical threshold ( $p_c^s(d)$ ), the authors must establish rigorous upper bounds for $p_c^b(d)$ , $p_c^s(d)$ , and $\beta_c(d)$ .
- They introduce a novel second-moment method involving "stationary random walks." Unlike previous methods that used purely oriented paths (which leave much of the space undetected), their strategy allows the path to perform a stationary random walk along one coordinate and a positive-oriented walk along the remaining $d-1$ coordinates.
- This approach reduces the intersection probability of two independent paths, leading to tighter upper bounds on critical probabilities for intermediate dimensions ( $5 \le d \le 10$ ).
- For $\beta_c(d)$ , they utilize infrared bounds from Fröhlich, Simon, and Spencer [11] and derive explicit upper bounds using properties of the modified Bessel function.

Key Contributions and Results

Theorem 1 (Bond Percolation): For each $d \ge 9$ , there exists $\epsilon_d > 0$ such that for $p \in (p_c^b(d), p_c^b(d) + \epsilon_d)$ , the complement of the infinite open cluster $X_p(d)$ stochastically dominates a supercritical site percolation on $\mathbb{Z}^d$ with parameter $\tilde{p} > p_c^s(d)$ .
- Improvement: This improves upon the $d \ge 10$ result of Bock et al. [4] and the $d \ge 11$ result of [9].
- Numerical Extension: The authors note that using numerical estimates for $p_c^b(d)$ and $p_c^s(d)$ (Table 3), the result likely holds for all $d \ge 4$ .
Theorem 2 (Ising Model): For the Ising model on $\mathbb{Z}^d$ with $d \ge 8$ , there exists $\epsilon_d > 0$ such that for $\beta \in [0, \beta_c(d) + \epsilon_d)$ , the $+$ spins in the minus phase stochastically dominate a supercritical site percolation.
- Improvement: This improves the result of Aizenman et al. [1], which required $d$ to be "large enough."
- Numerical Extension: Numerical estimates suggest this holds for all $d \ge 5$ .
Proposition 1 & 2: These propositions provide the core stochastic domination mechanisms. Proposition 1 shows that $V(\mathbb{Z}^d) \setminus C_\infty$ dominates site percolation with parameter $(1-p)^{2d}$ (a baseline bound). Proposition 2 establishes the domination for the Ising model based on free energy estimates.
Critical Probability Bounds: The paper provides rigorous upper bounds for $p_c^b(d)$ and $p_c^s(d)$ for $4 \le d \le 15$ (Table 2) using the new stationary random walk second-moment method. These bounds are sharper than existing literature for $5 \le d \le 10$ .

Significance and Claims
The paper claims to significantly lower the dimensional threshold for proving the coexistence of infinite clusters in both bond percolation and the Ising model. By shifting the focus to stochastic domination, the authors provide a stronger result than previous existence proofs.

The authors explicitly state that their method relies on the ability to bound critical probabilities effectively. They acknowledge that while their analytical method proves the result for $d \ge 9$ (percolation) and $d \ge 8$ (Ising), numerical evidence strongly suggests the coexistence holds for all $d \ge 4$ (percolation) and $d \ge 5$ (Ising).

Crucially, the authors are modest about the limitations of their method in low dimensions. They explicitly state in Remark 3 and Remark 4 that they believe their stochastic domination method fails for $d=3$ (and likely $d=4$ for Ising), even though they expect the physical phenomenon of coexistence to hold in those dimensions. They argue that in $d=3$ , the complement of the infinite cluster likely contains an infinite component but does not stochastically dominate a supercritical site percolation. Thus, the paper's contribution is the rigorous establishment of a specific, strong form of coexistence in high dimensions, rather than a proof of coexistence in all dimensions.

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