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.
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Notes
In the literature, it is the inverse of \(\alpha \) which is sometimes referred to as the Poincaré and Log-Sobolev constant.
The simulations in [3] covered also more general degree sequences.
Here and in the rest of the paper, the term “couple” refers to an ordered pair.
Indeed, assume that the edge \((i'_{ij},j'_{ij})\) does not satisfy condition 1. Then necessarily \((i'_{ij},j'_{ij})\) is a multiedge in \(G_1'\), so that \(i'_{ij}\in I(G_1,G_1')\), whereas, by the choice of \((i'_{ij}j'_{ij})\), \(\textbf{i2}(i'_{ij})=\textbf{i1}(i)\in I(G_1,G_1')\). Thus, either 2a or 2b must hold.
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Acknowledgements
The authors would like to thank Catherine Greenhill for bringing to their attention the paper [22]. This project was initiated when the first named author visited the second named author at New York University Abu Dhabi. Both authors would like to thank the institution for excellent working conditions. The first named author was partially supported by the Sloan Fellowship.
Funding
Funding is provided by Alfred P. Sloan Foundation (Grant No. FG-2019-11704).
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Tikhomirov, K., Youssef, P. Sharp Poincaré and log-Sobolev inequalities for the switch chain on regular bipartite graphs. Probab. Theory Relat. Fields 185, 89–184 (2023). https://doi.org/10.1007/s00440-022-01172-7
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DOI: https://doi.org/10.1007/s00440-022-01172-7
Keywords
- Switch chain
- Random regular graph
- Mixing/relaxation time
- Poincaré and log-Sobolev inequalities
Mathematics Subject Classification
- 60J10
- 05C80