Direct asymptotic analysis of coloured stochastic vertex models via their Plancherel spectral theory, and extension of the coloured formalism

The coloured (multi-species) stochastic vertex models built from the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_{n+1})$ possess a complete spectral theory: eigenfunctions of their transfer matrices are realized as lattice partition functions, with dual bases, biorthogonality, and Plancherel isomorphisms. Borodin and Wheeler, who constructed this theory, proved that all limit theorems for the coloured height function known at the time come only through a distributional matching with colour-blind ($n=1$) models, a route that requires process-level convergence of the uncoloured height function and is therefore rigorously available only in restricted regimes. They explicitly left as their central open direction the direct asymptotic analysis of the coloured models through the spectral (Plancherel) decomposition itself - including local-equilibrium, Gaussian-field, polymer/KPZ-equation, and KPZ-fixed-point regimes, and multi-point joint distributions of colours - together with extensions of the coloured formalism to inhomogeneous lattices, spin $q$-Whittaker functions, and elliptic/trigonometric IRF (interaction-round-a-face) models. Direct asymptotics for coloured models has so far been obtained only through observable/Cauchy-identity methods, LLT-integral asymptotics, probabilistic methods, and, most strongly, colored line ensembles: Aggarwal, Corwin, and Hegde proved that the colored ASEP and stochastic six-vertex model with fully packed initial data converge under KPZ scaling to the Airy sheet and directed landscape. None of these routes uses the Plancherel decomposition, which remains completely unexercised, and all three formalism extensions remain open in the coloured setting.

Background

A stochastic vertex model is a random system of up-right lattice paths on a quadrant in which the transition probabilities at each vertex (the vertex weights) are built from an $R$-matrix (a solution of the Yang-Baxter consistency equation) of a quantum group (a $q$-deformation of the symmetry algebra of a Lie algebra), so that the row-to-row transfer matrices commute and the model is exactly solvable. The stochastic $R$-matrix for the quantum affine algebra $U_q(A_n^{(1)})$ due to Kuniba, Mangazeev, Maruyama and Okado produces models with $n$ particle species, called colours: each path carries a colour in $\{1,\dots,n\}$, a vertex can absorb several paths of the same colour from below, and the weights favour lighter colours passing to the right of heavier ones. For $n=1$ these degenerate to the extensively studied colour-blind stochastic higher-spin six-vertex models, while degenerations of the coloured family give interacting particle systems such as the multi-species asymmetric simple exclusion process (ASEP, particles of different priorities hopping on a one-dimensional lattice with at most one particle per site) and the $q$-PushTASEP (a $q$-deformed pushing exclusion process, in which a jump pushes a block of neighbouring particles forward). The colour-resolved height function $H_{>c}(x,y)$ counts the paths of colour greater than $c$ that pass below the lattice point $(x,y)$; it is a random function of space, time, and the colour cut-off.

The monograph of Borodin and Wheeler (Astérisque 440) developed a full spectral theory for these coloured models: the eigenfunctions of the coloured transfer matrices were constructed as partition functions of the lattice model with specially chosen boundary conditions, together with a dual basis, biorthogonality relations, and Plancherel isomorphisms (a Plancherel formula is an identity that expands an arbitrary function or expectation over the eigenfunction basis, and is the standard analytic gateway from algebraic integrability to asymptotics). The same work also proved a matching theorem: the distribution of the coloured height function at a single point equals the distribution of the uncoloured height function along a line in an associated $\mathfrak{sl}_2$-related model. Since asymptotics of uncoloured models are well understood, the matching transfers limit theorems to coloured models, and Borodin and Wheeler spelled out four target asymptotic types for the coloured height function: (1) local-equilibrium (product-Bernoulli-type) limits, such as rarefaction-fan behaviour of coloured particles (the triangular space-time region in which the particle density interpolates between the left and right equilibrium densities); (2) Gaussian random-field fluctuations around the hydrodynamic limit (the deterministic large-scale density profile governed by a first-order PDE) for symmetric ($q=1$) or weakly asymmetric ($q\to 1$) scalings; (3) intermediate weakly asymmetric scalings converging to positive-temperature directed polymers (random paths in a random environment weighted by exponential path energies) and the Kardar-Parisi-Zhang (KPZ) equation, the nonlinear stochastic PDE of interface growth; (4) convergence, at fixed $q\ne 1$ or large weakly asymmetric scales, to the KPZ fixed point, the conjecturally universal scaling limit for asymmetric interface growth.

The matching route has a structural limitation that motivates the open problem: to extract a non-trivial coloured result from a one-point uncoloured one needs process-level convergence of the uncoloured height function, which is rigorously known only for a few determinantal models (models whose correlation functions are determinants and are hence exactly computable); essentially no coloured predictions existed beyond a few fluctuating-hydrodynamics conjectures of unclear applicability. Borodin and Wheeler therefore stated that it would be extremely interesting to access any of the four asymptotic regimes directly, through the spectral analysis (the Plancherel theory) of the coloured transfer matrices rather than the matching, and that such a route could yield stronger results - for example process-level coloured convergence, for which at the time no limiting object (a multi-species KPZ fixed point) had been constructed. In the same spirit they expressed the hope that the Cherednik-Dunkl operator technique (difference operators of the affine Hecke algebra, acting diagonally on Macdonald-type eigenfunctions, which they used to compute averages of coloured Hall-Littlewood observables) could be extended to understand joint distributions of colours under multiple lattice points, not just a single vertex.

Finally, they identified three natural generalizations of the coloured formalism that they deliberately left out of scope: (i) inhomogeneous lattice models, in which columns carry different spin parameters and rapidities; (ii) the spin $q$-Whittaker limit, where the coloured eigenfunction theory should be deformed to the spin $q$-Whittaker functions (one-parameter deformations of the $t=0$ Macdonald polynomials, a two-parameter family of symmetric polynomials diagonalized by commuting difference operators, introduced in the colour-blind case by Borodin and Wheeler); and (iii) elliptic and trigonometric IRF (interaction-round-a-face) or SOS (solid-on-solid) models, in which the Boltzmann weights attach to faces with height variables rather than to edges and obey the elliptic quantum group $E_{\tau,\eta}(\mathfrak{sl}_2)$ relations - colour-blind stochastic versions of these were built by Borodin. For context, colour-blind analogues of all three extensions exist in the literature, but the coloured (rank $n$) versions do not.

Aggarwal, Corwin, and Hegde (2024) proved, for two degenerations of the coloured family - the colored ASEP and the colored stochastic six-vertex model with fully packed initial data - that the two-parameter colored height function converges under KPZ scaling to the Airy sheet (the two-parameter random field whose space-time version, the directed landscape, is the established universal scaling limit of the KPZ class), with the $k$-colour space-time scaling limit identified as a marginal of the directed landscape. Their method embeds the colored models into colored Hall-Littlewood line ensembles (colour-indexed families of non-crossing random curves with an explicit Gibbs resampling property, under which a curve conditioned to stay between its two neighbours resamples as an independent non-crossing bridge), building on the colored line ensembles of Aggarwal and Borodin; the Plancherel spectral theory is not used. Process-level colored convergence therefore no longer awaits the spectral route for these two degenerations; what remains open is the spectral route itself, its application to the general coloured higher-spin vertex models that the line-ensemble and matching routes do not cover, the Gaussian-field and polymer regimes, and the multi-point and extension questions below.

Problem Statement

The overarching objective is to turn the Plancherel spectral theory of the coloured stochastic vertex models into an analytic machine that produces probabilistic limit theorems for these models directly, bypassing the distributional matching with colour-blind ($n=1$) models, and to extend the same spectral formalism to the natural generalizations of the coloured models.

(a) Direct spectral asymptotics. Derive asymptotic results for the coloured height function of the coloured stochastic vertex model in a quadrant (with domain-wall or half-Bernoulli boundary conditions: no paths enter from below; for the domain-wall (step) data one coloured path enters from the left in each row, with colours increasing with the row index, while for the half-Bernoulli data a particle of colour $i$ is placed at location $-i$, independently for each $i$, with probability $p\in(0,1]$, the case $p=1$ reducing to the step data) directly from the spectral decomposition of its transfer matrices - the eigenfunction expansion, biorthogonality, and Plancherel isomorphisms - rather than through the matching with colour-blind models. Following the source, it suffices to access any one of the four asymptotic regimes enumerated there directly through this spectral route: (1) local-equilibrium limits, such as the asymptotic behaviour of coloured particles in a rarefaction fan; (2) Gaussian random-field fluctuations around the hydrodynamic limit for symmetric or weakly asymmetric scalings; (3) intermediate weakly asymmetric scalings converging to positive-temperature directed polymers and the KPZ equation; (4) convergence at fixed $q\ne 1$ or large weakly asymmetric scales to the KPZ fixed point. Since the source was written, process-level KPZ convergence of the coloured height function has been proved for two degenerations of this family - the colored ASEP and the colored stochastic six-vertex model with fully packed initial data, converging to the Airy sheet - but by line-ensemble methods, not through the spectral theory; the distinctive open targets for the spectral route are therefore the general coloured higher-spin vertex models and boundary data not covered by the line-ensemble and matching routes, and results beyond what those routes can reach. A legitimate solution must make the Plancherel decomposition load-bearing in the derivation: results reachable only through the colour-blind matching, only through observable/Cauchy identities of the kind already in the literature, or only through line-ensemble (Gibbs-property) methods do not by themselves answer this question.

(b) Multi-point joint colour distributions. Characterize the joint distribution of the colour-resolved height function at multiple specified lattice points (equivalently, of the colours of paths passing under several specified locations), for example by extending the Cherednik-Dunkl operator technique or by spectral means, beyond the currently known multi-point $q$-moment observables and symmetry (shift- and flip-invariance) laws.

(c) Extension of the coloured spectral formalism. Carry out, in the coloured (rank $n$) setting, the construction of the eigenfunction bases with their biorthogonality and Plancherel formulae for: (i) inhomogeneous lattice models, in which columns carry different spin parameters and rapidities; (ii) the spin $q$-Whittaker limit, the coloured analogue of the colour-blind spin $q$-Whittaker polynomials; and (iii) elliptic and trigonometric IRF (interaction-round-a-face) or SOS models, the coloured analogues of the colour-blind elliptic stochastic face models, with the same stochasticity requirements on the weights.

The central open problem is (a): the source leaves to a future work the access to any one of the four enumerated asymptotic regimes through the spectral analysis of the coloured transfer matrices rather than the matching with colour-blind models, and a direct spectral derivation of any one of these regimes for the coloured height function, at the level of rigour of the source monograph, answers this problem. Items (b) and (c) are additional directions recorded by the source and are not requirements attached to (a): (b) appears in the source only as the hope that the Cherednik-Dunkl approach might be extendable to understanding joint distributions of colours of paths passing under multiple lattice locations, and (c) lists three generalizations that the authors called readily extendable but deliberately left out of the scope of their paper.

References

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