Observables of Stochastic Colored Vertex Models and Local Relation

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. Definition 2.2 Let Q ≤ P be a pair of up-left paths of length l, and let c = (c 1 , c 2 , . . . , c l ), z = (ζ 1 , ζ 2 , . . . , ζ l ) be parameters assigned to the steps of Q satisfying3 with the convention that the spectral parameter of a vertex is the ratio of the row and column rapidities.