The colored stochastic six-vertex model in a half space - the folded, boundary-reflected version of the flagship exactly solved lattice model of integrable probability - was shown by He to possess shift invariance: the joint law of height functions whose cuts cross the reflecting diagonal is invariant under shifting a block of cuts in the row or column direction together with compensating shifts of the rapidity parameters. In the full space the corresponding theory is complete: Borodin-Gorin-Wheeler's shift invariance is subsumed by Galashin's characterization theorem, resolving their conjecture that any transformation of the model preserving the marginal distribution of every height function automatically preserves all joint distributions. The half-space theory is open at exactly this point. The problem asks for the symmetry classification of the half-space colored six-vertex model: does marginal-preservation imply joint-preservation in the presence of the reflecting diagonal and signed colors, and which nontrivial shift-type invariances hold for height functions whose cuts are close to but do not cross the diagonal, where horizontal and vertical shifts are known to fail and the shift parallel to the diagonal is trivial? No proof or refutation of either question is known.

Background

The stochastic six-vertex model is a random assignment of arrows to the edges of the square lattice: at each vertex, two arrows enter from the left and from below and two leave to the right and upward, and the local arrow configuration is chosen among the six admissible patterns with prescribed weights, selected so that these local choices define a consistent global probability measure. In the colored (multi-species) version, arrows carry colors 1<⋯<n1<\dots<n, and arrows of larger colors overtake arrows of smaller ones; an inhomogeneous rapidity parameter - the spectral parameter on which the local vertex weights depend - is attached to each row and column. The model is a central exactly solved object of integrable probability. Its basic observables are the colored height functions H≥i(x,y)H^{\ge i}(x,y), which count the arrows of colors ≥i\ge i separating the point (x,y)(x,y) from the origin; the joint distributions of finitely many such height functions encode all multi-point information about the model. Two symmetry principles organize these joint distributions in the full plane. First, Borodin, Gorin and Wheeler discovered shift invariance: the joint law of a collection of height functions is unchanged when a block of cuts is shifted in the vertical (or horizontal) direction together with a compensating shift of the corresponding rapidity parameters, a statement with far-reaching consequences for polymers, last-passage percolation (LPP, the model of maximal-weight up-right paths across independent random weights), and interacting particle systems. Second, Galashin proved a conjecture of Borodin-Gorin-Wheeler that characterizes all such symmetries: any transformation of the model parameters and cut positions that preserves the marginal distribution of each individual height function automatically preserves all joint distributions. Related full-space symmetry statements for polymer and LPP models were obtained by Dauvergne using the Robinson-Schensted-Knuth (RSK) correspondence, a bijection between last-passage paths and tableaux that turns parameter symmetries into distributional ones. He's work transfers this circle of ideas to the half space. The half-space model lives on {(x,y)∈Z≥02:x≥y}\{(x,y)\in\mathbb{Z}_{\ge0}^2 : x\ge y\} and is obtained by folding the full-space colored model along the diagonal: the diagonal carries special reflecting boundary weights satisfying the reflection equation, the boundary counterpart of the Yang-Baxter consistency equation that underlies integrability, and the folding introduces negative (signed) colors on edges reflected off the boundary. He proved flip invariance (an auxiliary symmetry exchanging colors on the two sides of the diagonal) and a shift invariance theorem for this model: the joint law of height functions whose cuts all cross the diagonal is invariant under shifting a block of cuts by one row (or column) with compensating rapidity shifts. Via limits, fusion (combining several lattice rows or columns into single higher-spin vertices) and analytic continuation, analogous statements follow for the asymmetric simple exclusion process (ASEP, the one-dimensional interacting particle system in which particles hop asymmetrically subject to exclusion) on a half-line, for a fused and then continuous vertex model degenerating to the random walk in beta random environment of Barraquand and Rychnovsky, and for the absorption time of the type B oriented swap process, the analogue on signed permutations of the oriented swap process on permutations (the Markov process in which each adjacent pair of entries swaps exactly once, at a random time, so that the permutation is fully sorted at absorption) whose absorption-time asymptotics were found by Bufetov, Gorin and Romik, following the joint-identity conjecture of Bisi, Cunden, Gibbons and Romik that Lingfu Zhang proved in the full-space setting. What is missing is the symmetry classification itself. All of He's invariance theorems require the height functions' cuts to cross the diagonal. For height functions close to but not crossing the diagonal, horizontal and vertical shifts already fail to preserve the distribution of even a single height function in the inhomogeneous model, and the shift parallel to the diagonal is trivial, so no analogue of the full-space theory is known in the boundary regime. Moreover, no half-space analogue of Galashin's characterization (marginal-preservation implies joint-preservation) has been established, and the reflecting boundary and the signed colors break the spatial uniformity on which the full-space proof relies. The open problem is to determine the complete symmetry structure of the half-space colored six-vertex model.

Problem Statement

Consider the inhomogeneous colored stochastic six-vertex model in the half space {(x,y)∈Z≥02:x≥y}\{(x,y)\in\mathbb{Z}_{\ge0}^2 : x\ge y\} with integrable reflecting boundary weights, in the conventions of He's paper (signed colors on edges reflected off the diagonal, diagonal weights satisfying the reflection equation, arbitrary row and column rapidity parameters). Determine the complete symmetry structure of its height-function joint distributions, in two linked parts. (a) Characterization: let τ\tau be a transformation of the model data, namely the rapidity parameters together with the positions and color cutoffs of finitely many height-function observables (in He's conventions each height function H≥i(x,y)H^{\ge i}(x,y) carries a color cutoff ii in addition to its position, and the crossing-diagonal shift invariance transforms color cutoffs together with positions and rapidities), whose induced action preserves the marginal distribution of every individual height function. Prove that τ\tau then preserves the joint distribution of every finite collection of height functions (the half-space analogue of Galashin's full-space theorem), or disprove this by an explicit transformation that preserves all marginals while changing some joint law. (b) Non-crossing regime: for height functions whose cuts are close to but do not cross the diagonal, and more generally for an arbitrary finite collection of cuts, classify which nontrivial shift-type transformations of the rapidity parameters, cut positions and color cutoffs preserve the joint distribution. Here a shift-type transformation is one generated by the moves of He's crossing-diagonal shift invariance (Theorem 1.3): block shifts of the cut positions together with the compensating shifts of the rapidity parameters and color cutoffs. A transformation is nontrivial when it is neither the identity nor a shift parallel to the diagonal; any answer must account for the known obstructions that horizontal and vertical shifts fail to preserve even a single non-crossing height-function marginal in the inhomogeneous model and that the shift parallel to the diagonal is trivial. Parts (a) and (b) form one classification objective for this model: (b) supplies the boundary-regime data that any characterization theorem in (a) must incorporate, and (a) is the structural implication that makes the classification complete. Resolving the task requires both (a) and (b). Part (a) is settled either by a positive theorem, showing that marginal-preservation forces joint-preservation, or by an explicit counterexample, a transformation that preserves all marginals while changing some joint law. Part (b) requires an exhaustive classification of the nontrivial shift-type transformations that preserve the joint distribution, which a single exhibiting or obstructing example does not by itself settle, although the classification may combine positive and negative results.

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