Sharp Poincaré and log-Sobolev inequalities for the switch chain on regular bipartite graphs

Abstract

Consider the switch chain on the set of d-regular bipartite graphs on n vertices with $$3\le d\le n^{c}$$ , for a small universal constant $$c>0$$ . We prove that the chain satisfies a Poincaré inequality with a constant of order O(nd); moreover, when d is fixed, we establish a log-Sobolev inequality for the chain with a constant of order $$O_d(n\log n)$$ . We show that both results are optimal. The Poincaré inequality implies that in the regime $$3\le d\le n^c$$ the mixing time of the switch chain is at most $$O\big ((nd)^2 \log (nd)\big )$$ , improving on the previously known bound $$O\big ((nd)^{13} \log (nd)\big )$$ due to Kannan et al. (Rand Struct Algorithm 14(4):293–308, 1999) and $$O\big (n^7d^{18} \log (nd)\big )$$ obtained by Dyer et al. (Sampling hypergraphs with given degrees (preprint). arXiv:2006.12021 ). The log-Sobolev inequality that we establish for constant d implies a bound $$O(n\log ^2 n)$$ on the mixing time of the chain which, up to the $$\log n$$ factor, captures a conjectured optimal bound. Our proof strategy relies on building, for any fixed function on the set of d-regular bipartite simple graphs, an appropriate extension to a function on the set of multigraphs given by the configuration model. We then establish a comparison procedure with the well studied random transposition model in order to obtain the corresponding functional inequalities. While our method falls into a rich class of comparison techniques for Markov chains on different state spaces, the crucial feature of the method—dealing with chains with a large distortion between their stationary measures—is a novel addition to the theory.