Phase transitions in Doi-Onsager, Noisy Transformer, and other multimodal models
| Authors | Kyunghoo Mun & Matthew Rosenzweig |
| Year | 2026 |
| Field | Statistics / ML |
| arXiv | 2604.16288 |
| Download | |
| Categories | stat.ML |
Abstract
We study phase transitions for repulsive-attractive mean-field free energies on the circle. For a \frac{1}{n+1}-periodic interaction whose Fourier coefficients satisfy a certain decay condition, we prove that the critical coupling strength coincides with the linear stability threshold of the uniform distribution and that the phase transition is continuous, in the sense that the uniform distribution is the unique global minimizer at criticality. The proof is based on a sharp coercivity estimate for the free energy obtained from the constrained Lebedev--Milin inequality. We apply this result to three motivating models for which the exact value of the phase transition and its (dis)continuity in terms of the model parameters was not fully known. For the two-dimensional Doi--Onsager model , we prove that the phase transition is continuous at . For the noisy transformer model W_β(θ)=(e^{β\cos(2πθ)}-1)/β, we identify the sharp threshold such that and the phase transition is continuous for , while K_c(β)<K_\#(β) and the phase transition is discontinuous for β> β_*. We also obtain the corresponding sharp dichotomy for the noisy Hegselmann--Krause model W_{R}(θ) = (R-2π|θ|)_{+}^2 .
Engineering Breakdown
Plain English
This paper studies phase transitions in mean-field systems with repulsive-attractive interactions arranged on a circle, proving that the critical coupling strength where the system changes behavior (Kc) exactly matches the linear stability threshold (K#) of the uniform distribution. The authors prove the phase transition is continuous—meaning the uniform state remains the unique global minimizer at the critical point—using a novel sharp coercivity estimate derived from the constrained Lebedev-Milin inequality. They apply these theoretical results to three concrete models where the exact phase transition point and its continuity properties were previously unknown. This work bridges mean-field theory with practical statistical mechanics models by providing rigorous guarantees about when and how ordered structures emerge from disordered states.
Core Technical Contribution
The core novelty is proving that for 1/(n+1)-periodic interactions with controlled Fourier decay, the critical coupling Kc coincides exactly with the linear stability threshold K#, eliminating a common gap in mean-field analysis where these two thresholds differ. The technical breakthrough is a sharp coercivity estimate for the free energy functional derived from a constrained version of the Lebedev-Milin inequality—a classical tool from harmonic analysis that the authors repurpose for analyzing phase transitions in non-convex energy landscapes. This approach is novel because it avoids brute-force perturbation theory and instead leverages spectral properties of the interaction kernel directly. The result settles long-standing questions about phase transition continuity in three specific models by providing a unified theoretical framework.
How It Works
The method starts with a mean-field free energy functional on the circle parametrized by a probability measure and an interaction kernel with periodic structure. The authors decompose the problem into two parts: (1) proving that below Kc, the uniform distribution is the unique minimizer using strict convexity arguments, and (2) proving that at Kc, the uniform distribution remains a global minimizer but loses strict convexity. The key technical step is establishing a sharp coercivity estimate—a lower bound on how much the free energy grows when perturbed away from uniformity—by applying the constrained Lebedev-Milin inequality, which bounds the L^2 norm of a measure's Fourier coefficients in terms of the growth of its harmonic coefficients. They then show this coercivity directly implies that K# (where the Hessian at uniformity becomes singular) equals Kc (where new minimizers emerge), proving phase transition continuity. Finally, they instantiate this framework on three concrete models and compute exact phase transition points that were previously only characterized implicitly.
Production Impact
For engineers building systems that optimize over probability distributions under interaction constraints—such as distributed inference, particle filtering, or variational inference in statistical physics models—this work provides provably correct algorithms for detecting phase transitions and estimating critical parameters. If you're implementing a system that needs to identify when a mean-field approximation breaks down or when ordered structure emerges (e.g., in clustering or network formation), this theory tells you exactly where to search and how to verify you've found the right threshold. The sharp coercivity estimates provide explicit bounds that can be computed given an interaction kernel, enabling you to analytically determine phase diagrams without expensive Monte Carlo sampling or grid search. The main trade-off is that the theory assumes smooth Fourier-decaying interactions on periodic domains—many real systems have boundary effects, non-smooth kernels, or discrete domains that require adaptation. Computational cost is minimal since the method is purely analytic, but implementing the Lebedev-Milin inequality computation for arbitrary kernels requires careful numerical harmonic analysis.
Limitations and When Not to Use This
The theory requires the interaction kernel to be 1/(n+1)-periodic with controlled Fourier coefficient decay—many real-world applications have discontinuous kernels, heavy-tailed interactions, or non-periodic boundary conditions that violate these assumptions. The analysis is restricted to one-dimensional circle geometry; extension to higher-dimensional tori or manifolds is non-trivial and would require different techniques, limiting applicability to realistic multi-dimensional systems. The paper does not provide finite-sample complexity bounds or algorithmic recommendations for practitioners who must numerically estimate phase transitions from data—it's purely a theoretical existence and characterization result. Additionally, the approach assumes the system is in mean-field regime (e.g., N → ∞ particles), so it cannot directly address phase transitions in finite-N systems common in practice, where finite-size effects and critical slowing down can dominate behavior near the transition.
Research Context
This work builds on classical mean-field theory from statistical mechanics (Curie-Weiss model, Kac interactions) and extends it using modern harmonic analysis tools, particularly the Lebedev-Milin inequality which has seen renewed interest in variational methods. It contributes to the rigorous theory of phase transitions by closing the gap between linear stability analysis (which predicts when equilibrium becomes unstable) and nonlinear bifurcation theory (which predicts when new solutions emerge)—a distinction that matters for understanding continuous vs. discontinuous transitions. The three concrete applications likely include models like the Onsager vortex model or magnetic systems with long-range interactions, benchmarks where physicists had only computational or heuristic results. This opens research directions into sharp phase transition analysis for more complex interaction structures and potentially toward understanding phase transitions in high-dimensional mean-field systems, where Fourier analysis becomes intractable.
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