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Observables of Stochastic Colored Vertex Models and Local Relation

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Abstract

We study the stochastic colored six vertex (SC6V) model and its fusion. Our main result is an integral expression for natural observables of this model—joint q-moments of height functions. This generalises a recent result of Borodin–Wheeler. The key technical ingredient is a new relation of height functions of SC6V model in neighboring points. This relation is of independent interest; we refer to it as a local relation. As applications, we give a new proof of certain symmetries of height functions of SC6V model recently established by Borodin–Gorin–Wheeler and Galashin, and new formulas for joint moments of delayed partition functions of Beta polymer.

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Notes

  1. Throughout this work we will often treat 0 as an additional color denoting the absence of a path.

  2. In [BW18] and other related works the spectral parameter z in the weights \(R_z\) is the inverse of ours. This difference will be offset later, by setting spectral parameter of a vertex equal to the ratio x/y of the row and column rapidities, rather than y/x like in [BW18].

  3. with the convention that the spectral parameter of a vertex is the ratio of the row and column rapidities.

  4. The number of the incoming paths of a given color coincides with the number of the outgoing paths of a given color.

  5. Fusion in an algebraic setting was originally introduced in the works [KRS81] and [KR83].

  6. As before, the spectral parameter z of a vertex is the ratio of the row and the column rapidities.

  7. Following [BW20, Section 2.2], this can be readily seen from (2.8) by noticing that for \(z=1\) the sum over \({\varvec{P}}\) has only one nonzero term, at \({\varvec{P}}={\varvec{B}}\).

  8. These definitions are motivated by the desire to reorder a sequence of cuts in a way such that the points \({\mathfrak {q}}^{pre}_i\) go from the bottom-right endpoint of Q in the up-left direction and the points \({\mathfrak {p}}^{pre}_i\) go from the top-left endpoint of P in the down-right direction, but since achieving this is impossible in general, we distinguish two extreme cases which separately prioritize the order of the points \(\{{\mathfrak {q}}^{pre}_i\}\) or the order of the points \(\{{\mathfrak {p}}^{pre}_i\}\).

  9. The functions f used in [BW20] can be expressed using the operators \(T_\pi \), see [BW20, Remark 6.8].

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Acknowledgements

We are grateful to A. Borodin for many very helpful discussions. The work of A. Bufetov was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2047 “Hausdorff Center for Mathematics”. S. Korotkikh was partially supported by the NSF FRG grant DMS-1664619.

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  1. Hausdorff Center for Mathematics and Institute for Applied Mathematics, University of Bonn, Bonn, Germany

    Alexey Bufetov

  2. Massachusetts Institute of Technology, Cambridge, USA

    Sergei Korotkikh

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  1. Alexey Bufetov
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  2. Sergei Korotkikh
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Correspondence to Sergei Korotkikh.

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Communicated by H.Spohn.

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Bufetov, A., Korotkikh, S. Observables of Stochastic Colored Vertex Models and Local Relation. Commun. Math. Phys. 386, 1881–1936 (2021). https://doi.org/10.1007/s00220-021-04162-3

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