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

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 $(1+1\mathrm{/}n)(1-2\mathrm{/}n)$ 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 $n^2\mathrm{/}(n+2)$ and a mixing time of $O(n^2 \log n)$\mathrm{.}Weconjecturethatthemixingtimeisinfact$O(n \log n)$ as for the usual random-to-random shuffling.