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Title:
Multidimensional random walks and exactly solvable phase models
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Abstract:
The theory of random walks being the classical part of enumerative combinatorics appears in different fields of mathematics, physics and information theory. We consider random multi-dimensional lattice walks restricted by a hyperplane. This type of walks we call the walks over the multi-dimensional simplicial lattices. We demonstrate that the generating functions of these walks are the dynamical correlation functions of certain type of the exactly solvable quantum phase models describing strongly correlated bosons on a chain. The walks over the orientated lattices are related to the phase model with the non-Hermitian Hamiltonian while the walks over disorientated ones are related to the model with the Hermitian Hamiltonian. The calculation of the generating functions of random walks is based on the Quantum Inverse Method approach to the solution of the exactly solvable models. The answers are expressed through the symmetric functions. The realization of lattice paths in terms of step operators of the Phase model allows to represent them as a spin states and to calculate their entanglement. The continuous-time quantum walks bounded by one-dimensional lattice of finite length are also studied.
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