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Correlation comparisons and critical curves for disordered XY models

This paper establishes lower comparison bounds and regularity properties for disorder-averaged ferromagnetic XY models by leveraging Ginibre's inequality and Jensen-type arguments to analyze the effects of random couplings and site dilution on phase transitions and susceptibility thresholds in two dimensions.

Original authors: Yan Ru Pei

Published 2026-09-15

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CC BY 4.0

Original authors: 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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet world of statistical physics, researchers study how vast collections of tiny particles organize themselves. Imagine a sheet of material where every atom acts like a tiny compass needle, free to point in any direction. At high temperatures, these needles spin wildly and point in random directions, creating a state of disorder. But as the material cools, something remarkable happens: the needles begin to align with their neighbors, creating a unified order. In two dimensions, this transition is special. Instead of snapping into a rigid lock, the system enters a delicate "algebraic phase." Here, the needles do not point in exactly the same direction, but they remain correlated over vast distances, whispering to one another across the entire sheet. This state is fragile. If you remove some of the connections between the needles or scatter the atoms, the whispering might stop, and the order could vanish. The central question for decades has been whether this delicate phase can survive even a small amount of damage, and exactly how the temperature at which it breaks down shifts as the material becomes more imperfect.

A researcher has now answered these questions with mathematical certainty for a specific type of damaged material. They studied a model where the connections between the compass needles are either missing entirely or have random, fluctuating strengths. By developing a new way to compare a damaged system to a perfect one, they proved that the algebraic phase is incredibly robust. They showed that if the damage is weak, the phase survives at temperatures that are only slightly lower than in a perfect material. More importantly, they demonstrated that the relationship between the amount of damage and the temperature at which the phase breaks is smooth and predictable. There are no sudden jumps or chaotic surprises; as you gradually remove more connections, the critical temperature slides down in a steady, controlled manner.

The researcher achieved this by creating a mathematical bridge between a messy, random system and a clean, orderly one. They realized that the average effect of random, fluctuating connections could be replaced by a single, fixed connection that is slightly weaker. This replacement is not an approximation but a rigorous lower bound, meaning the real system is always at least as ordered as this simplified version. Using this insight, they could track how the system behaves as the density of missing links increases. They found that for both missing links and missing atoms, the temperature threshold where the algebraic phase disappears is a continuous function. It does not jump erratically; instead, it changes in a way that is locally smooth, meaning small changes in the amount of damage lead to proportionally small changes in the critical temperature.

This work also settled a specific question about the nature of the damage. Previous theories wondered if a tiny bit of randomness could completely destroy the phase even at temperatures where a perfect system would still be ordered. The new results prove that this is not the case. As long as the damage is sufficiently weak, the algebraic phase persists. The researcher also quantified how much the temperature drops when the connections have random noise. They found that if the noise is centered around an average value, the drop in temperature is proportional to the square of the noise's strength. This means that very small amounts of noise have a negligible effect, providing a "quadratic protection" that keeps the phase stable against minor fluctuations.

Furthermore, the study addressed how the system behaves in finite, real-world-sized samples rather than just in theoretical infinite sheets. They derived explicit lower bounds for the magnetic susceptibility, a measure of how easily the material responds to an external magnetic field. In the region where the algebraic phase exists, this response grows linearly with the size of the sample, confirming that the long-range correlations are indeed present and strong. This finding holds true even when the material is not perfectly uniform, provided the density of the remaining connections is high enough. The researcher also showed that the critical temperature remains stable even if the entire distribution of connection strengths changes slightly, as long as the maximum possible strength remains the same. This continuity suggests that the phase is a fundamental property of the system's structure, not an artifact of perfect uniformity.

The implications of these findings extend beyond the specific model studied. By proving that the critical curve is locally Lipschitz continuous, the researcher established that the transition from order to disorder is a smooth process, governed by clear mathematical rules. This smoothness implies that the system does not undergo sudden, unpredictable shifts as it is damaged. Instead, it degrades gracefully, with the temperature threshold sliding down in a predictable fashion. The work provides a solid foundation for understanding how disorder affects complex systems, offering a rigorous framework that can be applied to other materials and models. It confirms that the delicate algebraic phase is not a fragile illusion that vanishes at the first sign of imperfection, but a resilient state that can withstand a significant degree of disorder before finally giving way.

Technical Summary: Correlation Comparisons and Critical Curves for Disordered XY Models

Problem Statement
The paper addresses the stability of the algebraic phase in the two-dimensional ferromagnetic XY model under disorder, specifically bond and site dilution. While Dario and Garban [DG25] established the existence of an algebraic phase at sufficiently low temperatures on supercritical percolation clusters, the precise behavior of the critical temperature (or inverse temperature threshold) as a function of the disorder density remained open. Specifically, the work investigates:

Whether the algebraic phase survives weak dilution at temperatures strictly above the clean critical point ( $\beta_{BKT}$ ).
The regularity (continuity and differentiability) of the averaged critical curve $\beta_{c,\star}(p)$ with respect to the density of occupied bonds or sites $p$ .
The continuity of critical thresholds under perturbations of the entire coupling distribution, particularly regarding bounded noise and changes in the coupling law.

Methodology
The core methodology relies on a combination of probabilistic comparison techniques and correlation inequalities specific to the XY model.

Ginibre's Inequality and Transformed Convexity: The author utilizes Ginibre's correlation inequality, which ensures non-negativity of correlations for integer-charge cosine observables. A key technical innovation is the observation that normalized correlations are convex functions of the transformed coupling variable $H_\kappa(t) = 1 - e^{-2\kappa t}$ . This convexity allows the application of Jensen's inequality to disorder-averaged systems.
Effective Coupling Replacement: For independent nonnegative random couplings $K_e$ , the author proves that the disorder-averaged correlation is bounded below by a deterministic system with an "effective" coupling $K_e^{\text{eff}} = -\frac{1}{2} \log \mathbb{E}[e^{-2K_e}]$ . This transforms the problem of comparing disordered systems to comparing deterministic systems with modified coupling strengths.
Thinning and Star-Wise Jensen Passes:
- Bond Dilution: The comparison is achieved by "thinning" a higher-density Bernoulli graph to a lower-density one, replacing random couplings with their effective deterministic counterparts.
- Site Dilution: On bipartite graphs, the author employs a "two-pass" Jensen argument. By conditioning on one color class of the bipartition and averaging over the other, they handle the site dilution. This involves a star-wise application of Jensen's inequality, where the effective coupling is modified by a factor related to the maximum degree $\Delta$ of the graph (specifically using parameters $\alpha=2, \kappa=4$ for site dilution).
Finite-Domain Criteria: The paper leverages the clean finite-domain criterion of van Engelenburg and Lis [vEL23], which provides explicit lower bounds on correlations in finite boxes based on boundary sums, to transfer results from finite volumes to infinite-volume limits.

Key Results

Regularity of the Averaged Critical Curve (Theorem 1.1):
The paper proves that the averaged critical curve $\beta_{c,\star}(p)$ (defining the onset of algebraic decay) is finite, non-increasing, and locally Lipschitz continuous on the supercritical interval $(p_{c,\star}, 1]$ .
- The function $p \mapsto p^\alpha H_\kappa(\beta_{c,\star}(p))$ is non-decreasing.
- Explicit Lipschitz bounds are derived: $0 \le \beta_{c,\star}(p) - \beta_{c,\star}(q) \le \frac{\alpha e^{2\kappa M}}{2\kappa a}(q-p)$ for $a \le p \le q \le b$ , where $M = \beta_{c,\star}(a)$ .
- Parameters differ by model: $(\alpha, \kappa) = (1, 1)$ for bond dilution and $(2, 4)$ for site dilution.
Weak-Dilution Stability (Corollary 1.2):
The algebraic phase survives all sufficiently weak dilution at any inverse temperature $\beta > \beta_{BKT}$ . Specifically, if $p^\alpha H_\kappa(\beta) \ge H_\kappa(\beta_{BKT})$ , the averaged two-point function satisfies a polynomial lower bound $F_\star(p, \beta; x, y) \ge \frac{1}{8\|x-y\|_\infty}$ .
Susceptibility and Finite-Size Bounds (Section 5):
- The same regularity results apply to the susceptibility divergence threshold $\beta_{\chi,\star}(p)$ .
- Explicit lower bounds for magnetic susceptibility in finite boxes are derived, showing linear growth with the box side length in the weak-dilution region.
- Quadratic Protection: For bounded, centered bond noise ( $J_e = J_0 + \epsilon X_e$ ), the suppression of the algebraic-phase temperature is at most quadratic in the noise amplitude $\epsilon$ . The shift is bounded by $T_{alg}(\epsilon) \ge T_0 - C\epsilon^2$ .
Continuity under Coupling Law Perturbations (Theorem 6.1):
The critical threshold $c(\mu)$ is continuous with respect to the total variation distance between coupling laws $\mu$ and $\nu$ , provided the laws are supported on a common compact interval $[0, M]$ .
- If the reference law $\mu$ has an atom at the upper bound $M$ , the threshold is locally Lipschitz.
- If $\mu$ only has an essential supremum at $M$ (no atom), continuity still holds, though the Lipschitz constant may not be uniform without the atom.

Significance and Claims
The paper claims to resolve Open Questions 2–3 from Dario, van Engelenburg, and Garban [DvEG26] regarding the continuity of the critical curve for bond and site dilution. By establishing explicit comparison inequalities, the work demonstrates that the "temperature loss" due to dilution vanishes uniformly as the density loss vanishes.

The significance lies in providing a rigorous, quantitative framework for disorder in the XY model that goes beyond existence proofs. The author emphasizes that their comparisons are local and independent of volume, allowing for the derivation of Lipschitz regularity for the critical curve. Furthermore, the results on weak noise and coupling law perturbations provide the first explicit bounds on the robustness of the algebraic phase against small, bounded disorder variations, showing that the phase is stable against quadratic perturbations in noise amplitude.

The paper explicitly notes that it does not identify the critical curve with an exponential-decay threshold for the diluted model, nor does it prove the equality of the algebraic, susceptibility, and exponential decay thresholds in the disordered case, leaving these as open problems. The results are strictly derived from the averaged correlations and the phase-existence theorem of Dario and Garban.

Technical Summary: Correlation Comparisons and Critical Curves for Disordered XY Models

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