  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.
