Journal of Mathematical Sciences

, Volume 224, Issue 2, pp 199–213 | Cite as

Multi-Dimensional Random Walks and Integrable Phase Models

  • N. Bogoliubov
  • C. Malyshev

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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.St.Petersburg Department of Steklov Institute of MathematicsITMO UniversitySt.PetersburgRussia

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