Ungarian Markov Chains

TL;DR

This paper introduces the Ungarian Markov chain on finite lattices, analyzes its expected steps to reach the bottom, and provides asymptotic estimates and bounds for specific lattice classes using combinatorial and probabilistic methods.

Contribution

It defines the Ungarian Markov chain, establishes bounds for its expected hitting time, and connects lattice theory with last-passage percolation to derive new asymptotic estimates.

Findings

Asymptotic estimates for the expected steps on weak order and Tamari lattices.

Upper bounds for expected steps in Cambrian and $ u$-Tamari lattices.

Connection between lattice properties and last-passage percolation results.

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 $\nu$-Tamari lattices.