Multi-dimensional random walks and integrable phase models

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.