We consider random multi-dimensional lattice walks bounded by a hyperplane, calling them walks over multi-dimensional simplicial lattices. We demonstrate that generating functions of these walks are dynamical correlation functions of a certain type of exactly solvable quantum phase models describing strongly correlated bosons on a chain. Walks over oriented lattices are related to the phase model with a non-Hermitian Hamiltonian, while walks over disoriented ones are related to the model with a Hermitian Hamiltonian. The calculation of the generating functions is based on the algebraic Bethe Ansatz approach to the solution of integrable models. The answers are expressed through symmetric functions. Continuous-time quantum walks bounded by a onedimensional lattice of finite length are also studied. Bibliography: 40 titles.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 448, 2016, pp. 48–68.
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Bogoliubov, N., Malyshev, C. Multi-Dimensional Random Walks and Integrable Phase Models. J Math Sci 224, 199–213 (2017). https://doi.org/10.1007/s10958-017-3405-5
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DOI: https://doi.org/10.1007/s10958-017-3405-5