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Equality of the critical inverse temperatures for the one- and two-sided Dyson models

This paper proves that the critical inverse temperatures for the one-sided and two-sided Dyson models coincide when the interaction power $\alpha$ lies strictly between 1 and 2, while conjecturing that this equality also holds for the case $\alpha=2$ .

Original authors: Noam Berger, Anders Johansson, Anders Öberg

Published 2026-08-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Noam Berger, Anders Johansson, Anders Öberg

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

In the study of how matter changes its state, physicists often look at simple models to understand complex behaviors like magnetism. Imagine a long line of tiny magnets, each able to point either up or down. In the most basic version of this idea, each magnet only cares about its immediate neighbors. If they point in the same direction, they are happy; if they point in opposite directions, they are unhappy. In a one-dimensional line with only these immediate neighbors, the magnets never manage to align perfectly across the whole line, no matter how cold the system gets. They remain disordered. However, nature is rarely so simple. In the Dyson model, a specific variation of this problem, the magnets are allowed to feel the pull of neighbors far away, not just the ones touching them. The strength of this long-distance pull weakens as the distance grows, following a specific rule. When this pull is strong enough but still fades quickly enough, the system behaves differently: it can suddenly snap into a unified, ordered state, much like a magnet becoming magnetic. This sudden change is called a phase transition, and it happens at a specific temperature threshold.

A key question arises when we change the shape of the world these magnets inhabit. Does it matter if the line of magnets stretches infinitely in both directions, or if it starts at a point and goes on forever in just one direction? Intuitively, having a "wall" at one end of the line might make it harder for the magnets to coordinate their alignment across the whole system. This would suggest that the temperature required to trigger the phase transition is higher for the one-sided line than for the two-sided line. For decades, this inequality was the best guess, but a definitive proof that the two thresholds are actually the same remained elusive for a specific range of interaction strengths.

In a new paper, researchers Noam Berger, Anders Johansson, and Anders Öberg have settled this question for a wide range of interaction strengths. They proved that for the Dyson model, the critical temperature at which the phase transition occurs is exactly the same whether the system is a line stretching infinitely in both directions or a line starting at a point and stretching infinitely in one direction. This result holds true whenever the long-range pull between magnets fades at a rate that is neither too slow nor too fast. The team did not rely on computer simulations or approximations; they constructed a rigorous mathematical proof that demonstrates the two critical points must coincide.

The researchers approached the problem by looking at the system through a different lens, using a method that translates the magnetic spins into a network of connections. Instead of thinking about up and down arrows, they imagined a graph where connections between points appear or disappear based on the temperature. In this view, the phase transition happens when a giant, infinite cluster of connected points suddenly forms. The team knew that if the two-sided system has such a giant cluster, the one-sided system must also have one, because removing part of the line can only make connections harder, not easier. The difficult part was proving the reverse: that if the one-sided system has a giant cluster, the two-sided system must have one too. To do this, they had to show that the "harder" one-sided system is actually just as capable of forming a giant cluster as the "easier" two-sided system.

Their strategy involved a clever technique called renormalization, which is essentially a way of zooming out to see the big picture. They divided the line of magnets into large blocks. They showed that if the temperature is high enough to cause a phase transition in the two-sided system, then within any large block of the one-sided system, there is a very high probability of finding a large, internally connected cluster of magnets. These clusters act like solid islands of order. The researchers then demonstrated that these islands, even in the one-sided system, are likely to connect to each other across the gaps between the blocks. They used a method they call "sprinkling," which involves adding a tiny amount of extra randomness to the connections between these blocks. This small addition is enough to bridge the gaps and ensure that the islands merge into a single, infinite network.

The proof relies on the specific way the interaction strength fades with distance. The researchers found that their method works perfectly when the interaction exponent is between one and two. In this range, the long-range connections are strong enough to link the distant blocks together, even in the presence of the boundary. The logic flows from the existence of large internal clusters to the formation of a spanning network, and finally to the conclusion that the critical temperature for the one-sided system cannot be higher than that of the two-sided system. Since it was already known that the one-sided temperature cannot be lower, the two must be equal.

There is one case that the paper does not resolve. When the interaction strength fades at a rate corresponding to an exponent of exactly two, the mathematical tools used in the proof no longer hold up. The researchers explicitly state that this borderline case remains an open problem. They suspect the equality still holds there, but their current argument cannot reach it. For all other cases where the interaction is long-range but summable, however, the mystery is solved. The presence of a boundary does not shift the temperature at which the system orders itself.

This finding is significant for the broader field of statistical physics because it confirms that the fundamental nature of the phase transition is robust against changes in geometry, provided the interactions are sufficiently long-range. It also validates a specific approach used in other areas of physics, such as the study of transfer operators, where the behavior of one-sided systems is often analyzed. By proving that the critical threshold is the same, the authors show that the full-line behavior is the natural reference point even for systems with a boundary. The work stands as a precise mathematical confirmation that, in this specific world of long-range magnets, the edge of the line does not change the rules of the game.

Technical Summary: Equality of Critical Inverse Temperatures for One- and Two-Sided Dyson Models

Problem Statement
The paper addresses the Dyson model, a one-dimensional Ising model with long-range pair interactions decaying as $J(ij) = |i - j|^{-\alpha}$ . The study focuses on the regime $1 < \alpha \le 2$ , where interactions are summable yet sufficiently long-range to induce a phase transition at low temperatures, defying standard one-dimensional nearest-neighbor intuition.

The central problem is the comparison of critical inverse temperatures ( $\beta_c$ ) between two geometries:

The Full Line ( $\mathbb{Z}$ ): The two-sided model.
The Half-Line ( $\mathbb{N}$ ): The one-sided model.

In the FK-Ising random-cluster representation, deleting the negative half-line restricts connectivity, implying that the half-line critical inverse temperature $\beta^{\mathbb{N}}_c$ is at least as large as the full-line value $\beta^{\mathbb{Z}}_c$ (i.e., $\beta^{\mathbb{Z}}_c \le \beta^{\mathbb{N}}_c$ ). The paper aims to prove the reverse inequality, thereby establishing equality $\beta^{\mathbb{N}}_c = \beta^{\mathbb{Z}}_c$ for the range $1 < \alpha < 2$ .

Methodology
The proof relies on the FK-Ising random-cluster representation and employs a combination of stochastic domination, sprinkling techniques, and block renormalization. The core strategy involves demonstrating that percolation on the half-line occurs whenever the system is above the full-line critical point.

Stochastic Domination and Sprinkling:
The authors utilize the domain Markov property to establish that the free-boundary measure on a larger interval stochastically dominates the measure on a sub-interval. They employ a "sprinkling" argument, adding an independent Bernoulli graph with edge probabilities $1 - e^{-\delta J(ij)}$ to the FK-Ising configuration. This allows the comparison of the measure $\nu^{\mathbb{N}}_\beta$ with a slightly higher temperature measure $\nu^{\mathbb{N}}_{\beta+2\delta}$ , effectively bridging the gap between the full-line and half-line behaviors.
Finite-Volume Cluster Estimates:
A key technical component is Lemma 2, which establishes that above the full-line critical point $\beta^{\mathbb{Z}}_c$ , large finite intervals contain internal clusters of size at least $N^\gamma$ with high probability, provided $\alpha/2 < \gamma < 1$ . This estimate is derived via an inductive renormalization scheme.
Block Renormalization and Percolation Reduction:
The proof constructs a coarse-grained model by partitioning the half-line into blocks of length $N$ .
- Internal Clusters: Within each block, the existence of a large internal cluster is guaranteed by the finite-volume estimate.
- Inter-Block Connections: Independent long-range connections (sprinkling) are added between these large clusters.
- Reduction to Site-Bond Percolation: The connectivity of these blocks is shown to dominate a one-dimensional long-range site-bond percolation model. The site density corresponds to the probability of a block containing a large cluster, and the bond probabilities are derived from the long-range interaction strengths between blocks.
Application of Newman-Schulman Theorem:
The authors invoke the theorem of Newman and Schulman regarding long-range percolation with exponent $\alpha < 2$ . This theorem guarantees percolation in the coarse-grained model if the site density and bond probabilities are sufficiently high. By choosing the block size $N$ large enough, the authors ensure the dominating model is supercritical.

Key Contributions and Results

Main Theorem: The paper proves that for the Dyson model with interaction exponent $1 < \alpha < 2$ , the critical inverse temperatures for the one-sided and two-sided models coincide: $\beta^{\mathbb{N}}_c = \beta^{\mathbb{Z}}_c$ .
Extension to Random-Cluster Models: The argument is shown to extend to the general random-cluster model with cluster weight $q \ge 1$ , implying $\beta^{\mathbb{N},q}_c = \beta^{\mathbb{Z},q}_c$ for the same range of $\alpha$ .
Technical Bound: The proof relies on the strict inequality $2\gamma - \alpha > 0$ (where $\gamma < 1$ is the cluster size exponent). This condition is necessary for the renormalization step to yield a supercritical percolation model.

Significance and Limitations

Motivation: The result resolves a question regarding the transfer-operator approach to one-sided Dyson potentials. Previous work by Johansson, Öberg, and Pollicott established the existence of continuous eigenfunctions in the subcritical regime $\beta < \beta^{\mathbb{Z}}_c$ (for $\alpha > 3/2$ ). The equality $\beta^{\mathbb{N}}_c = \beta^{\mathbb{Z}}_c$ confirms that the full-line threshold is indeed the natural threshold for the one-sided system in the range $1 < \alpha < 2$ .
Open Problem: The authors explicitly state that their argument fails for the borderline case $\alpha = 2$ because the condition $2\gamma - \alpha > 0$ cannot be satisfied with $\gamma < 1$ . Consequently, the equality $\beta^{\mathbb{N}}_c = \beta^{\mathbb{Z}}_c$ for $\alpha = 2$ remains an open problem, though the authors conjecture it to be true.
Modesty: The paper does not claim to solve the $\alpha=2$ case or provide new applications beyond the theoretical unification of critical thresholds. It strictly confines its claims to the proof of equality for $1 < \alpha < 2$ using the specific renormalization and sprinkling framework described.

Technical Summary: Equality of Critical Inverse Temperatures for One- and Two-Sided Dyson Models

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