Fluctuation theory for the Ehrenfest urn

The Ehrenfest urn model with d balls, or alternatively random walk on the unit cube in d dimensions, is considered in discrete and continuous time, together with related models. Attention is focused on the fluctuation theory of the model-behaviour on unusual states-and in particular on first passage to the opposite vertex. Applications to statistical mechanics, reliability theory and genetics are surveyed, and some new results are obtained.