  Diaconis and Sturmfels introduced an influential method to construct Markov
chains using commutative algebra. One major point of their method is that
infinite families of graphs are simultaneously proved to be connected by a
single algebraic calculation. For large state spaces in the infinite families
these Markov chains are not rapidly mixing and only ad hoc methods have been
available to improve their mixing times. We provide a method to get rapid
mixing by constructing expanders for the Diaconis-Sturmfels type Markov chains.
