Ungarian Markov Chains

Abstract

We introduce the Ungarian Markov chain ${\bf U}_L$ associated to a finite lattice $L$. The states of this Markov chain are the elements of $L$. When the chain is in a state $x\in L$, it transitions to the meet of $\{x\}\cup T$, where $T$ is a random subset of the set of elements covered by $x$. We focus on estimating $\mathcal E(L)$, the expected number of steps of ${\bf U}_L$ needed to get from the top element of $L$ to the bottom element of $L$. Using direct combinatorial arguments, we provide asymptotic estimates when $L$ is the weak order on the symmetric group $S_n$ and when $L$ is the $n$-th Tamari lattice. When $L$ is distributive, the Markov chain ${\bf U}_L$ is equivalent to an instance of the well-studied random process known as last-passage percolation with geometric weights. One of our main results states that if $L$ is a trim lattice, then $\mathcal E(L)\leq\mathcal E(\text{spine}(L))$, where $\text{spine}(L)$ is a specific distributive sublattice of $L$ called the spine of $L$. Combining this lattice-theoretic theorem with known results about last-passage percolation yields a powerful method for proving upper bounds for $\mathcal E(L)$ when $L$ is trim. We apply this method to obtain uniform asymptotic upper bounds for the expected number of steps in the Ungarian Markov chains of Cambrian lattices of classical types and the Ungarian Markov chains of $ν$-Tamari lattices.