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No infinite cluster at criticality for long-range percolation with fast-decaying interactions

This paper proves that independent long-range bond percolation on $\mathbb{Z}^d$ ( $d \ge 2$ ) with fast-decaying interaction kernels exhibits no infinite cluster at the critical parameter, thereby establishing the continuity of the phase transition for a broad class of models including power-law decays (except at the critical exponent $\alpha=d$ ) and exponentially decaying kernels.

Original author: Yan Ru Pei

Published 2026-11-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original author: Yan Ru Pei

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, invisible grid stretching out in every direction, like the streets of a city that never ends. On this grid, points are connected by invisible bridges. Sometimes these bridges appear by chance, linking a point to its immediate neighbor, but sometimes they can stretch across great distances, connecting a point to one far away. This is the world of long-range percolation, a mathematical model used by scientists to understand how connections form in complex systems, from the spread of diseases to the flow of electricity in a network. The central mystery in this field is a question of continuity: as we gradually increase the number of bridges, does the system suddenly snap into a state where a single, unbroken path spans the entire infinite grid, or does this transition happen smoothly? For decades, mathematicians have known that for short, local connections, this transition is smooth. But for connections that can jump across the grid, the answer has remained elusive, especially when those long jumps become rare but still possible.

A new study by Yan Ru Pei settles this question for a wide class of these long-range systems. The research proves that when the probability of a long jump drops off quickly enough as the distance increases, the transition to a giant, infinite cluster is always smooth. In other words, at the exact tipping point where an infinite cluster first becomes possible, it does not yet exist; it only appears once the connection strength is increased just a tiny bit further. This finding closes a significant gap in our understanding, confirming that the behavior of these systems with fast-decaying long-range links is fundamentally similar to the simpler, short-range models that have been studied for much longer.

The problem the researchers tackled is deceptively simple to visualize but incredibly difficult to prove. They considered a grid where any two points can be connected, but the likelihood of a connection depends on how far apart they are. If the points are close, the connection is likely; if they are far, it is unlikely. The specific condition for this study was that the chance of a connection must fade away very rapidly as distance grows. The researchers wanted to know what happens at the critical moment when the system is just strong enough to potentially form an infinite path. In many physical systems, such a moment is marked by a sudden, violent change, where a giant cluster appears out of nowhere. However, the team demonstrated that for these specific long-range models, no such sudden appearance occurs. At the critical threshold, the largest cluster remains finite, no matter how large the grid is.

To reach this conclusion, the author had to overcome a unique challenge posed by the long jumps. In standard models where connections are only between neighbors, a path leaving a specific area must cross the boundary of that area. This makes it easy to track and control the flow of connections. But in a system with long jumps, a path can simply leap over the boundary, appearing on the other side without ever touching the edge. This ability to "jump over walls" breaks many of the traditional tools used to analyze these systems. The researchers had to develop a new way of thinking about how clusters grow and interact. Instead of counting how many points on the edge of a region are connected, they measured the total "weight" of all the potential connections crossing that boundary. This allowed them to account for the possibility of long jumps without getting lost in the complexity of the grid.

The proof involves a sophisticated strategy of building connections layer by layer, moving from small, manageable blocks to larger and larger structures. The researchers imagined a process where they try to link a central point to the outside world. They showed that even if a cluster manages to escape a small box, the probability of it successfully reaching a distant target without getting stuck is extremely low unless the connection strength is well above the critical point. They used a technique that involves checking many different paths simultaneously, ensuring that if a cluster tries to escape, it will almost certainly be blocked by the lack of strong enough connections. By carefully controlling the size of the jumps and the density of the connections, they proved that the system cannot sustain an infinite cluster right at the critical moment.

This result is significant because it unifies our understanding of how networks behave. Previously, mathematicians knew that if the long jumps were very heavy and frequent, the transition could be abrupt and discontinuous. They also knew that for very short, local connections, the transition was smooth. The new study fills the middle ground, proving that as long as the long jumps become rare fast enough, the system behaves smoothly. This covers a vast range of scenarios, including systems where the connection probability drops off exponentially with distance, which is a common pattern in many real-world phenomena. The work also clarifies the boundary between smooth and abrupt transitions, showing that the smooth behavior holds true in all dimensions greater than one, provided the long-range interactions decay sufficiently quickly.

The researchers did not rely on computer simulations or approximations; their work is a rigorous mathematical proof. They showed that under the stated conditions, the probability of finding an infinite cluster at the critical point is exactly zero. This means that the phase transition, the shift from a disconnected world to a connected one, happens gradually. There is no sudden jump. The finding resolves a long-standing question in the field and provides a solid foundation for understanding how long-range interactions influence the structure of complex networks. It confirms that the intuition built on simple, local models holds true even when long-distance connections are allowed, as long as those connections are sufficiently rare.

The study also highlights the limits of this smooth behavior. The researchers noted that if the long jumps do not fade away quickly enough, the system can behave differently, potentially leading to a sudden, discontinuous transition. This distinction is crucial, as it defines the precise conditions under which networks evolve smoothly versus those where they undergo a sudden collapse or formation of a giant component. By pinpointing the exact mathematical conditions that guarantee a smooth transition, the work offers a clear map for future research and applications in network theory.

In the end, the paper provides a definitive answer to a question that has puzzled mathematicians for years. It shows that the universe of long-range percolation is more orderly than previously thought, at least when the long jumps are rare. The ability to leap across the grid does not necessarily lead to chaos or sudden changes; instead, the system maintains a gentle, continuous evolution toward connectivity. This insight deepens our understanding of how complex systems organize themselves and offers a powerful tool for predicting the behavior of networks in the real world, from social connections to the spread of information. The proof stands as a testament to the power of careful mathematical reasoning in unraveling the mysteries of complex systems, turning a seemingly intractable problem into a clear and elegant solution.

Technical Summary: No Infinite Cluster at Criticality for Long-Range Percolation with Fast-Decaying Interactions

Problem Statement
The paper addresses the continuity of the phase transition in independent long-range bond percolation on the integer lattice $\mathbb{Z}^d$ ( $d \ge 2$ ). In this model, distinct vertices $x$ and $y$ are connected with probability $p_{xy}(\beta) = 1 - e^{-\beta J(x-y)}$ , where $J$ is a symmetric, non-negative kernel. The central question is whether the percolation probability $\theta(\beta) = \mathbb{P}_\beta(0 \leftrightarrow \infty)$ vanishes at the critical parameter $\beta_c = \inf\{\beta : \theta(\beta) > 0\}$ .

While the continuity of the phase transition ( $\theta(\beta_c) = 0$ ) is established for nearest-neighbor percolation in $d=2$ and high dimensions, the long-range case depends critically on the decay rate of the kernel $J$ . Previous results covered "heavy-tailed" kernels (where $J(x) \sim \|x\|^{-d-\alpha}$ with $\alpha < d$ ) and specific fast-decaying cases (nearest-neighbor, finite-range, or high-dimensional spread-out kernels). The gap remained for general fast-decaying kernels in all dimensions $d \ge 2$ , particularly for power-law decays where $\alpha > d$ and for exponentially decaying kernels.

Methodology
The proof adapts a renormalization scheme originally developed for nearest-neighbor percolation on $\mathbb{Z}^3$ by Kozma and Nitzan, further refined in recent work on finite graphs. The argument proceeds by contradiction: assuming $\theta(\beta_c) > 0$ , the author constructs an infinite cluster at a parameter $q < \beta_c$ , which is impossible.

The proof consists of four main steps, modified to handle long-range edges:

Faces at a Percolating Parameter: The author establishes that a large box $\Lambda_m$ connects with high probability to "outer thick quarter-faces" of a surrounding cube $\Lambda_\ell$ . To handle long edges that might jump over the boundary, they introduce "thick" faces (slabs of width $w$ ) and utilize the tail condition on $J$ to bound the probability of edges longer than $w$ .
Extension through a Fresh Region: This step generalizes the "seed-and-exploration" method. If a cluster outside a box $B$ $B$ reaches $B$ $B$ , it must reach a target set $T$ $T$ with high probability. The author replaces the nearest-neighbor counting of boundary neighbors with a "crossing weight" $W$ $W$ , defined as the sum of edge weights between the external cluster and the box.
- If $W$ is small, the probability that all crossing edges are closed is high ( $e^{-W}$ ), making disconnection "cheap" but rare across layers.
- If $W$ is large, the landing points of the crossing edges support many independent attempts to open "seed boxes" inside the target region.
- Long edges jumping over the shell around the box are controlled by the tail condition.
Relay: A sequence of extension steps moves a connection from the center of one coarse cube to a neighboring one. This involves a series of contraction and translation steps between nested rectangles, ensuring the connection propagates through the lattice.
Exploration and Contour Count: The author performs an exploration of a coarse lattice $\mathbb{Z}^2$ embedded in $\mathbb{Z}^d$ . They define "good" coarse vertices based on the success of the relay steps. Using a planar contour counting argument (Lemma 2.3), they show that if the set of good vertices is finite, the probability of such a configuration decays exponentially. This implies that with positive probability, an infinite cluster of good vertices exists, leading to an infinite open cluster at parameter $q < \beta_c$ , a contradiction.

Key Contributions and Results
The primary result is Theorem 1.1, which states that for $d \ge 2$ , if the kernel $J$ satisfies:

(Sym): Invariance under coordinate permutations and sign changes.
(Pos): $J(e_1) > 0$ (positive nearest-neighbor interaction).
(Tail): $\liminf_{n \to \infty} n^d \sum_{\|x\|>n} J(x) = 0$ .

Then $0 < \beta_c < \infty$ and $\mathbb{P}_{\beta_c}(0 \leftrightarrow \infty) = 0$ .

This result yields two immediate corollaries:

Corollary 1.2: For power-law kernels $J(x) \asymp \|x\|^{-d-\alpha}$ with $\alpha > 0$ and $\alpha \neq d$ , the phase transition is continuous. (The case $\alpha < d$ was known via Berger's theorem; this paper covers $\alpha > d$ ).
Corollary 1.3: For exponentially decaying kernels (and finite-range kernels) satisfying the symmetry and positivity conditions, the phase transition is continuous.

Significance and Scope
The paper claims to settle the continuity of the phase transition for long-range percolation in all dimensions $d \ge 2$ for kernels that decay sufficiently fast (specifically satisfying the tail condition). It unifies previous results that were restricted to $d=2$ (planar arguments) or high dimensions (lace expansion).

The proof is notable for adapting the Kozma-Nitzan renormalization argument to long-range interactions. The author explicitly notes that long edges introduce two specific challenges:

Paths can jump over the surfaces that nearest-neighbor paths must cross.
Clusters outside a box can enter from far away without passing through the immediate boundary.

These are controlled respectively by the "thick outer faces" and the "crossing weight" measurement.

Limitations and Open Cases
The paper explicitly excludes the borderline case where $J(x) \asymp \|x\|^{-2d}$ (i.e., $\alpha = d$ ). In this regime, the tail condition fails ( $n^d T(n)$ does not vanish), and long edges of length comparable to $n$ leave the cube $[-n, n]^d$ with probability of order one at every scale. The author notes that for $d=1$ with $J(x) \asymp |x|^{-2}$ , the transition is known to be discontinuous. Whether $\theta(\beta_c) = 0$ holds for $\alpha = d \ge 2$ remains an open problem, as the current argument cannot guarantee the necessary scale separation.

Technical Summary: No Infinite Cluster at Criticality for Long-Range Percolation with Fast-Decaying Interactions

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