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Uniqueness for DLR equations of $\mathsf{Sine}_\beta$

This paper establishes the uniqueness of stationary solutions to the DLR equations for the unit-intensity $\mathsf{Sine}_\beta$ point process under a finite electric energy condition, thereby providing a canonical statistical physics characterization of the process and resolving a question posed by Dereudre, Hardy, Leblé, and Maïda.

Original authors: Theodoros Assiotis

Published 2026-09-09

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CC BY 4.0

Original authors: Theodoros Assiotis

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, invisible world of mathematics, there exists a class of problems that ask a simple question: if you have a vast collection of points scattered across a line, and those points push and pull on each other with a specific, long-range force, what does the final, stable arrangement look like? This is the realm of statistical physics, a field that seeks to understand how the chaotic behavior of individual particles gives rise to the predictable laws of the macroscopic world. For decades, scientists have studied a particular type of point system known as a "log gas," where the particles repel one another with a force that grows stronger the closer they get, similar to how electric charges interact. Among these systems, a specific pattern called the Sine process has emerged as a universal standard. It appears not just in abstract math, but as a description of the zeros of the Riemann zeta function, a famous object in number theory, and as the local behavior of eigenvalues in random matrices. The Sine process is the "gold standard" for how these points should settle when the system is in perfect equilibrium.

However, a fundamental question remained unanswered: is this Sine process the only way these points can arrange themselves to satisfy the rules of equilibrium? In physics, equilibrium is often described by a set of rules called the DLR equations. These rules state that if you look at any small, finite region of the system, the arrangement of points inside that region must be determined by the points outside it, in a very specific, balanced way. For many years, it was known that the Sine process follows these rules. But it was not known if other, stranger arrangements could also follow the same rules. If such other arrangements existed, it would mean the Sine process is not unique, and our understanding of this universal pattern would be incomplete. The question was whether the Sine process is the sole architect of this equilibrium, or just one of many possible builders.

A mathematician named Theodoros Assiotis has now provided a definitive answer to this question. In a rigorous proof, he demonstrated that for any strength of interaction between the particles, the Sine process is indeed the unique solution to these equilibrium rules, provided the system obeys one crucial condition: the total "electric energy" of the arrangement must be finite. This energy condition essentially ensures that the points are distributed evenly enough over the long run, preventing them from clumping together in a way that would create infinite strain. Assiotis proved that if you have a system that follows the equilibrium rules and has this finite energy, it must be the Sine process. There is no other possibility.

The path to this proof was not a straight line of calculation but a clever comparison of two different worlds. Assiotis took a hypothetical system that followed the equilibrium rules and compared it to a known, well-behaved system called the circular beta ensemble. This known system consists of points arranged on a circle, which makes the mathematics of their interactions much easier to handle. The core of the argument involved shrinking the infinite line down to a finite interval and then stretching it to match the size of the circle. By doing this, Assiotis could place the unknown system and the known circular system side by side. He then measured the "distance" between them, not in physical space, but in terms of information theory. This distance, known as relative entropy, quantifies how different two probability distributions are.

The breakthrough came from showing that as the interval grew larger and larger, this distance between the unknown system and the circular system shrank to nothing. It was not enough to show they were close; Assiotis had to show that the difference vanished completely in the limit. He achieved this by carefully tracking the energy contributions from the "exterior" of the interval—the points outside the region being studied. He proved that the influence of these distant points on the local arrangement becomes negligible, provided the system has finite energy. This allowed him to conclude that the local behavior of the unknown system is identical to the local behavior of the circular system. Since the circular system is known to converge to the Sine process, the unknown system must also be the Sine process.

The paper also clarifies what happens if the energy condition is dropped. Without the requirement of finite energy, the uniqueness of the Sine process collapses. Assiotis constructed specific examples of systems that follow the equilibrium rules but have infinite energy. These systems are essentially compressed or stretched versions of the Sine process, or even empty spaces, and they have different densities of points. These counterexamples show that the finite energy condition is not just a technical detail but a necessary physical constraint that forces the system into the unique, universal Sine pattern. Without it, the rules of equilibrium are too loose to pin down a single solution.

This work resolves a question that had been open for some time, settling a conjecture made by other researchers in the field. It provides a clean, statistical-physics definition of the Sine process, characterizing it not just by how it is built, but by how it behaves in equilibrium. The proof relies on a deep understanding of how local interactions and global constraints work together. By comparing the unknown to the known and showing that the gap between them disappears, the paper establishes that the Sine process is the only stable, finite-energy arrangement for this class of point systems. It is a result that reinforces the universality of the Sine process, confirming that in the grand balance of these interacting points, there is only one true equilibrium.

Technical Summary: Uniqueness for DLR Equations of Sine $\beta$

Problem Statement
The paper addresses the uniqueness of the stationary solution to the Dobrushin–Lanford–Ruelle (DLR) equations for the Sine $\beta$ point process, a universal bulk scaling limit for one-dimensional $\beta$ -log gases. While the Sine $\beta$ process was previously characterized via stochastic differential equations, explicit stochastic operators, and as the weak limit of circular $\beta$ -ensembles, its characterization via statistical mechanics remained incomplete. Specifically, Dereudre, Hardy, Leblé, and Maïda (DHLM) had established that Sine $\beta$ satisfies the canonical DLR equations and is number-rigid, but the question of whether Sine $\beta$ is the unique stationary solution to these equations (under a finite energy condition) remained open.

The central challenge is that the classical Gibbs variational principle, which typically links DLR solutions to free-energy minimizers, does not directly apply to the singular, long-range logarithmic interaction of the log-gas in a straightforward manner. Furthermore, for $\beta \neq 2$ , the determinantal structure and Riemann–Hilbert methods used to prove uniqueness for the $\beta=2$ case are unavailable.

Methodology
The author adopts a direct comparison strategy between a general stationary DLR solution and a suitably matched circular $\beta$ -ensemble, bypassing the free-energy functional entirely. The proof architecture proceeds through the following technical steps:

Number Rigidity and Conditioning: Leveraging the number rigidity property (established by DHLM), the conditional law of a DLR solution $P$ in a finite interval $I_L = [0, L]$ , given the exterior configuration $\xi$ , is supported on a single fiber with a fixed number of particles $n = n_L(\xi)$ .
Neutralization and Dilation: The conditional configuration is "neutralized" by dilating the interval $I_L$ to $I_n$ (length $n$ ) via the map $T_{n/L}$ . This transforms the conditional law into a measure $\nu_{L,\xi}$ on a fixed-particle-number fiber with a neutral charge distribution.
Reference Measure: This neutralized law is compared to the circular $\beta$ -ensemble $Q_{n,\beta}$ on a circle of circumference $n$ , which is also a Gibbs measure on the same fiber.
Relative Entropy Estimate: The core technical achievement is proving that the averaged relative entropy between the neutralized DLR conditional law and the circular ensemble is sub-extensive:
$H^{\text{circ}}_L = \int \text{Ent}(\nu_{L,\xi} \mid Q_{n_L(\xi),\beta}) \, \alpha_L(d\xi) = o(L).$
This estimate is derived by decomposing the difference in potentials into four components:
- Affine Dilation Statistic: Controlled using the sublinear discrepancy estimate $v(L) = o(L)$ derived from the finite renormalized electric energy condition.
- Periodic-to-Intrinsic Error: The difference between the periodic circular energy and the intrinsic interval energy is shown to be small under both laws.
- Exterior Boundary Term: A gauge-invariant boundary potential difference is controlled using exact neutrality and Stieltjes integration by parts, showing the exterior interaction error is $o(L)$ .
- Gibbs Comparison: Jensen's inequality is applied to the difference of potentials to bound the entropy.
Cyclic-Gap Transport: Since the entropy bound is $o(L)$ rather than $o(1)$ , Pinsker's inequality cannot directly imply convergence in total variation. Instead, the author employs a cyclic averaging argument. The entropy cost is distributed over the $n \approx L$ gaps between particles. Using transport inequalities (specifically Bregman divergences associated with the circular gap potential), it is shown that the expected cost per gap is $o(1)$ .
Palm Identification: The cyclic-gap transport establishes that the rooted gap laws of the candidate DLR solution converge to those of the circular ensemble. By invoking the known local limit of the circular ensemble (which converges to the reduced Palm law of Sine $\beta$ ) and the convergence of the candidate's rooted laws to its own reduced Palm law, the author concludes that the reduced Palm laws of the candidate and Sine $\beta$ coincide.
Palm Inversion: Finally, since both processes are stationary, simple, and have intensity one, the equality of their reduced Palm laws implies the equality of the processes themselves.

Key Contributions and Results

Theorem 1.1: For any $\beta > 0$ , the Sine $\beta$ point process is the unique Borel probability measure on the configuration space that is stationary, satisfies the canonical $\beta$ -DLR equations, and has finite process-level divergence-compatible renormalized electric energy ( $W_{\text{div}}(P) < \infty$ ).
Counterexamples: The paper demonstrates that the finite-energy condition is indispensable. By applying compression maps to Sine $\beta$ , the author constructs stationary DLR solutions with intensities $m \neq 1$ (and thus infinite renormalized energy). Furthermore, mixtures of such processes with the vacuum state can yield stationary, simple, intensity-one DLR solutions that are not Sine $\beta$ , provided they violate the finite-energy condition.
Technical Innovation: The paper introduces a new method for comparing Gibbs measures with singular logarithmic interactions in one dimension. Unlike previous approaches relying on determinantal structures or variational principles, this method relies on a direct relative entropy comparison between a DLR solution and a circular ensemble, utilizing exact neutrality and sublinear discrepancy estimates.

Significance
The paper provides a canonical statistical physics characterization of the Sine $\beta$ point process, resolving a question posed by DHLM. It establishes that Sine $\beta$ is not merely a solution to the DLR equations but the unique stationary solution within the class of finite-energy processes. This result solidifies the connection between the microscopic log-gas models and their macroscopic limits through the DLR formalism. The author notes that this characterization may be useful for proving convergence to Sine $\beta$ in pre-limit models with a natural Gibbs structure, though the paper does not pursue specific applications or extensions to higher dimensions, noting that the argument is essentially one-dimensional. The work also clarifies the role of the finite renormalized electric energy condition as a necessary constraint for uniqueness in the presence of long-range interactions.

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