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arXiv · 1412.7488

Spectral gap for random-to-random shuffling on linear extensions

Arvind Ayyer·Anne Schilling·Nicolas M. Thiéry
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Abstract

In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-random shuffling.

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Arvind Ayyer, Anne Schilling, Nicolas M. Thiéry. 2015-10-13. Spectral gap for random-to-random shuffling on linear extensions. https://doi.org/10.1080/10586458.2015.1107868

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