Corpus ID: 220363826

# Sharp Poincar\'e and log-Sobolev inequalities for the switch chain on regular bipartite graphs

@article{Tikhomirov2020SharpPA,
title={Sharp Poincar\'e and log-Sobolev inequalities for the switch chain on regular bipartite graphs},
author={Konstantin E. Tikhomirov and Pierre Youssef},
journal={arXiv: Probability},
year={2020}
}
• Published 6 July 2020
• Mathematics
• arXiv: Probability
Consider the switch chain on the set of $d$-regular bipartite graphs on $n$ vertices with $3\leq d\leq n^{c}$, for a small universal constant $c>0$. We prove that the chain satisfies a Poincare 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 Poincare inequality implies that in the regime $3\leq d\leq n^c$ the mixing time of the switch… Expand
3 Citations

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#### References

SHOWING 1-10 OF 61 REFERENCES
Simple Markov-chain algorithms for generating bipartite graphs and tournaments
• Mathematics, Computer Science
• SODA '97
• 1997
A simple Markov chain has one state for every graph (or bipartite graph) with the given degree sequence; in particular, there are no auxiliary states as in the chain used by Jerrum and Sinclair. Expand
Conductance and the rapid mixing property for Markov chains: the approximation of permanent resolved
• Mathematics, Computer Science
• STOC '88
• 1988
The permanent function arises naturally in a number of fields, including algebra, combinatorial enumeration and the physical sciences, and has been an object of study by mathematicians for many years (see [14] for background). Expand
Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences
• Computer Science, Mathematics
• SODA 2019
• 2018
This work resolves an open problem posed by Greenhill and Sfragara (2017) regarding the switch chain on undirected graphs and relies on a comparison argument with a Markov chain defined by Jerrum and Sinclair for sampling graphs that almost have a given degree sequence. Expand
Subgraphs of random graphs with specified degrees
If a graph is chosen uniformly at random from all the graphs with a given degree sequence, what can be said about its subgraphs? The same can be asked of bipartite graphs, equivalently 0-1 matrices.Expand
Sampling regular graphs and a peer-to-peer network
• Mathematics, Computer Science
• SODA '05
• 2005
A related Markov chain for d-regular graphs on a varying number of vertices is introduced and it is proved that the related chain has mixing time which is bounded by a polynomial in N, the expected number of Vertices, under reasonable assumptions about the arrival and departure process. Expand
COMPARISON THEOREMS FOR REVERSIBLE MARKOV CHAINS
• Mathematics
• 1993
By symmetry, P has eigenvalues 1 = I03 > I381 > ?> I 31xI- 1 2 -1. This paper develops methods for getting upper and lower bounds on 8i3 by comparison with a second reversible chain on the same stateExpand
The mixing time of switch Markov chains: A unified approach
• Computer Science, Mathematics
• Eur. J. Comb.
• 2022
This paper shows that on any P -stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing, and is an almost uniform sampler for power-law and heavy-tailed degree sequences. Expand
Sampling hypergraphs with given degrees
• Computer Science, Mathematics
• Discret. Math.
• 2021
This work describes and analyzes a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence, and gives some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a constant. Expand