REM universality and Poisson-Dirichlet Gibbs weights for linear random energy

Abstract: We study the Hamiltonian $H_n(h,σ)=\sum_{i=1}^n h_i(σ_i-m), $ where $(h_i)$ are i.i.d.\ real random variables and $(σ_i)$ are i.i.d.\ Ising spins. We consider the energy levels obtained after an independent thinning that retains an exponential number of configurations ($e^{O(n)}$). We prove that, after an $(h_i)$-dependent centering, the resulting point process converges in distribution to a Poisson point process with exponential intensity. Thus, the energy levels asymptotically has the one of the Random Energy Model (REM). Our results extend previous ones, where REM universality for this model was established only either for energy fluctuations of order $e^{-O(n)}$ or for $e^{o(\sqrt n)}$ randomly selected configurations. We also identify the limiting Gibbs weights, which converge to a Poisson--Dirichlet law, and the quenched free energy, which exhibits a freezing transition at $β=\tildeλ$. The proofs are presented here in compressed form; full details are given in the companion preprint.

• The paper establishes REM universality in a linear random energy model by applying an exponential thinning mechanism to the spin configurations.
• The paper proves the convergence of energy level point processes to a Poisson process with an exponential intensity, validating REM-like statistics.
• The paper demonstrates that in the low-temperature phase, Gibbs weights converge to a Poisson–Dirichlet distribution, reflecting a 1RSB freezing transition.

REM Universality and Poisson–Dirichlet Gibbs Weights in Linear Random Energy Models

Summary and Context

The paper "REM universality and Poisson–Dirichlet Gibbs weights for linear random energy" (2606.07757) investigates the emergence of Random Energy Model (REM) universality and associated Poisson–Dirichlet statistics in a purely linear random Hamiltonian subject to exponential thinning of the state space. Specifically, it analyzes the joint asymptotic statistics of energy levels and Gibbs weights for the model

$H_n(h, \sigma) = \sum_{i=1}^n h_i (\sigma_i - m),$

where the $h_i$ are IID real random variables and the $\sigma_i$ are independent Ising spins with prescribed magnetization $m \in (-1,1)$.

This work significantly advances earlier results that established REM universality only in exponentially shrinking energy windows or under very sparse selections ($e^{o(\sqrt n)}$) of the state space, often with an additional independent REM term. Here, REM universality is demonstrated for exponentially large ($e^{O(n)}$) randomly selected configurations in a strictly linear setting, without additional noise.

Results on Point Processes: Poisson Convergence

The primary mathematical result is that, following an appropriate centering of energies dependent on the environment $h$, the point process of retained energy levels converges in distribution to a Poisson point process with exponential intensity. This realization is expressed as

$$\mathbf H_n := \sum_{\sigma \in \mathcal{G}_n(U)} \delta_{H_n(h, \sigma) - A_n(h)} \xrightarrow{d} PPP(\mathcal{D}_{\tilde\lambda}),$$

where $\mathcal{G}_n(U)$ denotes the set of retained configurations, $A_n(h)$ is a deterministic centering given by a conditional large deviation analysis, and the limiting intensity $h_i$0 is fully characterized in terms of the underlying law of $h_i$1.

This convergence confirms that, despite the strong correlations introduced by the linear structure, the asymptotics of extreme and substantial fractions of the spectrum reproduce those of the REM—a model with fully independent energy levels.

Gibbs Weights: Poisson–Dirichlet and 1RSB Structure

In the low-temperature phase, i.e., for $h_i$2, the ranked Gibbs weights converge in law to the Poisson–Dirichlet distribution $h_i$3. This corresponds precisely to Derrida's 1RSB Ruelle probability cascade, which is central to the Parisi theory of mean-field spin glasses. Mathematically, for the weights

$h_i$4

the joint law of the order statistics converges in law to the Poisson–Dirichlet structure associated with the limiting point process. This regime is sharply separated by a freezing transition at $h_i$5, beyond which the Gibbs measure is dominated by a random, hierarchically organized set of pure states, as predicted by replica symmetry breaking.

Free Energy and Phase Transition

The paper delivers a rigorous computation of the explicit limiting free energy per spin:

$h_i$6

where $h_i$7 is a function determined by the quenched large deviation principle, and $h_i$8 are explicitly characterized by the energy statistics of the thinned process. The transition at $h_i$9 is of freezing type, with discontinuity in the second derivative of $\sigma_i$0, paralleling the precise phase structure observed in the original REM and signaling the emergence of a Poisson–Dirichlet regime.

Technical Approach

• Thinning Mechanism: The independent thinning retains an exponential ($\sigma_i$1) subset of Ising configurations, with each configuration selected with a probability proportional to a bias dependent on $\sigma_i$2.
• Large Deviation Framework: Detailed quenched large deviation estimates yield the conditional distribution and centering necessary for Poisson process convergence.
• Laplace Functional Method: Limit theorems for Laplace functionals and careful truncation analysis establish the convergence of point processes and Gibbs weights.
• Variance and Second Moment Arguments: The analysis of the free energy utilizes second moment and large deviation controls to show precise concentration of the normalization and the entropy profile.

Theoretical and Practical Implications

The work demonstrates that REM statistics, including the Poissonian nature of the energy spectrum and 1RSB thermodynamics (Poisson–Dirichlet Gibbs weights), emerge robustly under minimal assumptions—linear models with strong correlations—once sufficient (exponentially large) random thinning is applied. This substantiates conjectures that REM universality is far more general than initially anticipated and does not rely on independence or mixture with an independent REM term.

Practically, these results underpin simplified spin glass and neural memory models, with implications for associative memory capacity bounds and retrieval thresholds in high-capacity Hopfield-type networks. The explicit link to cross-attention architectures points to potential statistical mechanics interpretations of contemporary neural network components.

Future Directions

• Beyond Linear Models: Investigation of REM universality in more general correlated Hamiltonians, including non-linear and inhomogeneous energy landscapes.
• Finite-Size Corrections: Detailed quantitative analysis of finite-$\sigma_i$3 corrections and rate of convergence for applications in computational models.
• Algorithmic Implications: Exploitation of Poisson–Dirichlet statistics in algorithmic random sampling, variational inference, or optimization landscapes for high-dimensional systems.
• Connections to Deep Learnability: Further bridging between thermodynamic phase transitions in spin glasses and transition phenomena in large-scale neural networks.

Conclusion

The paper rigorously establishes REM universality and the emergence of Poisson–Dirichlet Gibbs weights and associated freezing transition in the context of linearly correlated random energies under exponential thinning. This broadens the domain of validity for REM-type asymptotics and 1RSB thermodynamics to a wider class of disordered systems, offering both novel theoretical insight and direct relevance to models of associative memory and deep learning (2606.07757).