Continuous time multidimensional walks as an integrable model

Continuous time walks in multidimensional symplectical lattices are considered. It is shown that the generating functions of random walks and the transition amplitudes of continuous time quantum walks are expressed through the dynamical correlation functions of the exactly solvable model describing strongly correlated bosons on a chain, the so-called phase model. The number of random lattice paths of fixed number of steps connecting the starting and ending points of the multidimensional lattice is expressed through the solutions of Bethe equations of the phase model. Its asymptotic is obtained in the limit of the large number of steps.