Expanders from Markov bases

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.