We prove that the switch chain on d-regular bipartite graphs on n vertices satisfies the Poincare inequality with a constant of order O(nd) provided that d is less than a small constant power of n; moreover, when d is fixed, we establish a log-Sobolev inequality for the chain with a constant of order O(n log(n)). We show that both results are optimal. The estimates allow to significantly improve known bounds on the mixing time of the chain.
Joint work with Pierre Youssef.