Schur rational functions, vertex models, and random domino tilings
Location
It is known that Schur symmetric polynomials admit a number of generalizations (Macdonald's 1992 variations) which retain determinantal structure - for example, factorial and supersymmetric Schur functions. We describe an overarching family of Schur-like rational functions arising as partition functions of fully inhomogeneous free fermion six vertex model. These functions are indexed by partitions, have as variables the pairs (x_i,r_i), i=1,...,N, of horizontal rapidities and spin parameters; and, moreover, depend on vertical rapidities and spin parameters (y_j,s_j), j>=1. We establish determinantal formulas, orthogonality, Cauchy identities, and other properties of our functions. We also introduce random domino tiling models based on the Schur rational functions (a la Schur processes of Okounkov-Reshetikhin 2001), and obtain bulk (lattice) asymptotics leading to a new deformation of the extended discrete sine kernel. Based on the joint project https://arxiv.org/abs/2109.06718 with A. Aggarwal, A. Borodin, and M. Wheeler.
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