Rowmotion Markov Chains

Rowmotion is a certain well-studied bijective operator on the distributive lattice $J(P)$ of order ideals of a finite poset $P$. We introduce the rowmotion Markov chain ${\bf M}_{J(P)}$ by assigning a probability $p_x$ to each $x\in P$ and using these probabilities to insert randomness into the original definition of rowmotion. Barnard and Thomas--Williams extended the definition of rowmotion from distributive lattices to semidistributive lattices and trim lattices, respectively. Recently, the first author and Williams further generalized rowmotion to the even broader class of semidistrim lattices. Given a semidistrim lattice $L$, we assign a probability $p_j$ to each join-irreducible element $j$ of $L$ and use these probabilities to construct a rowmotion Markov chain ${\bf M}_L$. Under the assumption that each probability $p_j$ is strictly between $0$ and $1$, we prove that ${\bf M}_{L}$ is irreducible. We bound the mixing time of ${\bf M}_{L}$ for an arbitrary semidistrim lattice $L$, and we provide more precise bounds when $L$ is a chain or a Boolean lattice. We compute the stationary distribution of ${\bf M}_{J(P)}$ for an arbitrary finite poset $P$. We also compute the stationary distribution of the rowmotion Markov chain of the lattice obtained from a disjoint union of two chains $x_1<\cdots<x_a$ and $y_1<\cdots