Geodesic
Multi-dimensional random walks and integrable phase models
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 48-68

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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 one-dimensional lattice of finite length are also studied. 
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     author = {N. Bogoliubov and C. Malyshev},
     title = {Multi-dimensional random walks and integrable phase models},
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     volume = {448},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a2/}
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N. Bogoliubov; C. Malyshev. Multi-dimensional random walks and integrable phase models. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 48-68. http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a2/