Correlated and uncorrelated long--time asymptotics of type D ASEP

Abstract: The type D ASEP is an asymmetric two--species interacting particle system on $\Z$, in which two separately conserved species hop, bind into a composite ``bound pair'', and split. The model, along with its reversible measures and orthogonal polynomial duality, was constructed using the representation theory of $U_q(\so_{2n})$. The reversible measures and orthogonal polynomial duality are each a product of two copies of the single-species ASEP reversible measures and orthogonal polynomial duality. In this paper, we study the long-time asymptotics of the type D ASEP. In the fixed--$q$ regime, using an exact current--decoupling identity, we prove that the asymptotic hydrodynamic limit and Tracy--Widom fluctuations decouple, as predicted from the duality. In the weak--asymmetry (Edwards--Wilkinson) regime, when $q=1-c/N<sup>2$, we prove that the two density fluctuation fields \underline{decouple}: each converges to a linear stochastic heat equation, with no cross--coupling in either the drift or the noise, the limiting noises having vanishing cross--correlation. More surprisingly, we then prove that the two limiting normal random variables are \underline{correlated} with a seemingly new correlation function. The correlation is exactly equal to $(1-e<sup>{-4c})/(4c)$, with the positive parts of the normal random variables having correlations expressed by the Bessel--Struve function. This paper, with the exception of the abstract and introduction, was written entirely by Claude Opus 4.8 and Fable 5. The proofs were then formalized in Lean, using Aristotle by Harmonic AI. The human author of this paper verified the proofs manually.

• The paper demonstrates that Type D ASEP achieves exact decoupling in hydrodynamic and fluctuation regimes, with species-specific currents independent of cross-derivative influences.
• Utilizing orthogonal polynomial duality and rigorous martingale analysis, the study proves convergence to independent stochastic heat equations in the EW regime and Tracy–Widom distributions in the KPZ regime.
• The research quantifies initial-condition induced correlations and validates its results through automated formal proofs and extensive numerical simulations.

Decoupling and Asymptotic Behavior in Type D ASEP

Model Construction and Structural Decoupling

Type D ASEP is a two-species asymmetric exclusion process defined on the integer lattice ℤ, where particles may hop, bind into composite "bound pairs," and split. The model’s generator, reversible product measure, and duality functions are derived from the representation theory of quantum groups of type D (𝑈q(𝔰𝔬{2n})), yielding a process with two separately conserved species exhibiting nontrivial interaction via binding, yet preserving a factorized reversible measure and orthogonal polynomial dualities. Notably, the reversible measure and the orthogonal polynomial duality decompose as products over two single-species counterparts, leading to the expectation of autonomous or decoupled hydrodynamics.

A salient structural feature emerges: the species–i current, both at the microscopic and macroscopic scale, depends solely on the occupancy of that species. All cross derivatives in the flux vanish, and the hydrodynamic Hessian is strictly diagonal. Thus, inter-species coupling manifests only through binding/splitting transitions, confined to the noise (fluctuations) and never to the drift; equilibrium cross-transport coefficients, both static and dynamic, are identically zero. This eliminates off-diagonal transport phenomena, producing decoupled hydrodynamics for each species.

Exact Decoupling in Fluctuation Regimes

The analysis addresses both the Edwards–Wilkinson (EW) and Kardar–Parisi–Zhang (KPZ) universality classes, treating asymptotic regimes with (A) weak asymmetry ($q = 1 - c/N$) and (B) fixed asymmetry ($q$ constant).

EW Regime: Utilizing the orthogonal Krawtchouk self-duality, a martingale problem analysis demonstrates that, under diffusive scaling, each fluctuation field converges to an independent linear stochastic heat equation with species-dependent noise. The two fields—density fluctuations of species 1 and 2—are uncorrelated in both drift and noise in the limit. Strong orthogonality statements ensure the cross-noise is entirely supported on two-body dual particle configurations; detailed duality reductions and kernel bounds provide explicit quantitative estimates. Tightness and uniqueness are established via formalized criteria (Aldous-Mitoma), verified in Lean.

KPZ Regime: Marginal fluctuations are governed by Tracy–Widom laws for each species, confirmed for all $n$ via autonomous single-species reductions. The joint current fluctuations decouple at the level of $q$-Laplace observables, with all inter-species coupling confined to contact events (binding). The joint covariance between the two species decays sublinearly ($s^{-1/6}$), conjectured to vanish asymptotically—supported by exact numerical simulations. Explicit contact kernel representations are derived, with rigorous bounds on the occupation times of the relative coordinate.

Residual Initial-Condition Driven Correlations

Despite decoupling in dynamics, initial conditions can induce persistent cross-species correlations, quantitatively analyzed in the block-bound pair regime. The limiting joint law of rescaled dual positions after a splitting event is a mixture of bivariate normals with correlation $\rho(c) = (1 - e^{-4c})/(4c)$, strictly between 0 and 1, decaying as $1/c$ for large $c$. The positive parts of the limiting random variables exhibit correlations expressible via the Bessel–Struve function, representing a novel functional form not previously documented in multi-species exclusion literature. These closed-form expressions are backed by analytic proofs and extensive Monte Carlo simulations.

Mathematical Formalization and Verification

All results, including martingale decompositions, orthogonality, kernel bounds, fluctuation convergence, and tightness criteria, are formalized in Lean 4 using automated theorem-proving (Aristotle CLI), with comprehensive human validation. Classical results such as the Nash inequality, Kolmogorov–Rogozin anti-concentration, Karamata's Tauberian theorem, and Aldous-Mitoma tightness are newly formalized for the project. A human-readable blueprint and dependency graph are available, supporting full computational reproducibility.

Implications and Future Directions

The structural exactness and rigorous decoupling in Type D ASEP position it uniquely among multi-species models. Unlike other colored ASEPs with genuine off-diagonal transport (e.g., stirring models, distinct-velocity KPZ), Type D's coupling is strictly a transient, initial-condition-driven artifact, not a transport effect. The model occupies an umbilic point in universality—the characteristic velocities coincide, eliminating mechanisms for non-diagonal hydrodynamics. The interplay between binding events and noise-driven correlation exemplifies a regime not previously described: EW-class, diagonal compressibility, vanishing cross-mobility, but with nonzero shared-source initial correlations.

Numerical evidence corroborates the rigorous results across parameter regimes and phase boundaries, confirming scaling exponents and asymptotic decoupling. Open questions remain on the joint $q$-moment tower, higher cumulants, and non-stationary EW analysis, especially regarding initial-condition cross-correlations beyond the block-bound regime.

Conclusion

Type D ASEP demonstrates a powerful structural decoupling in two-species exclusion processes, derived from underlying algebraic dualities and quantum group symmetries. Both hydrodynamic and fluctuation limits exhibit autonomous behavior, with all cross-species coupling confined to noise and initial conditions. The rigorous analysis, supported by formal proof and numerical simulation, establishes a novel universality class, with broad implications for the study of multi-species stochastic processes and non-equilibrium hydrodynamics in systems with composite binding. The framework, accessible and reproducible via formalization tools, provides a foundation for further investigation into higher-order correlations, exact duality constructions, and extensions to other quantum group symmetric interacting particle systems.

Reference: "Correlated and uncorrelated long--time asymptotics of type D ASEP" (2607.11376)