The problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions is considered. The model is formulated as a scalar product of off-shell Bethe states. The quantum inverse scattering method is applied, and three different integral representations for these states are obtained. By suitably combining such representations, and using certain antisymmetrization relation in two sets of variables, it is possible to derive integral representations for various correlation functions. In particular, focusing on the emptiness formation probability, besides reproducing the known result, obtained by other means elsewhere, a new one is obtained. By construction, the two representations differ in the number of integrations and their equivalence is related to a hierarchy of highly nontrivial identities.
https://doi.org/10.1016/j.nuclphysb.2021.115535
We consider the skew Howe duality for the action of certain dual pairs of Lie groups on the exterior algebra of the product of fundamental representations as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations, in particular, for the exterior algebra of the fundamental representation of the GL series and for the spinor representation in the SO series. In the limit of infinite group rank and infinite tensor power, we have obtained the limit shape of Young diagrams, which parametrize the irreducible representations included in the decomposition. We show that the limit shape is described by one analytical formula for all classical series. Uniform convergence to the limit shape is proved. It is shown that the limit shape and fluctuations around it are described by the Krawtchouk ensemble.
https://arxiv.org/abs/2111.12426
We derive a path counting formula for two-dimensional lattice path model with filter restrictions in the presence of long steps, source and target points of which are situated near the filters. The concept of congruence of connected regions in models of paths on a lattice is introduced. It turns out to be useful for deriving explicit formulas for calculating paths in an auxiliary path model in the presence of long steps, the beginning and end of which lie in filters. The problem is motivated by the fact that the weighted numbers of paths of such a model reproduce the multiplicities in the expansion of the tensor power of the two-dimensional U_q (sl_2)-module in roots of unity. The combinatorial properties of this model were studied, and a plan for the proof of the derivation of explicit formulas for calculating paths was outlined.
http://ftp.pdmi.ras.ru/pub/publicat/znsl/v509/p201.pdf
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