Tighter Bounds on the Expected Absorbing Time of Ungarian Markov Chains

TL;DR

This paper establishes new bounds on the expected absorption time of Ungarian Markov chains across different lattices, confirming conjectures and revealing linear and near-linear behaviors in specific cases.

Contribution

It provides the first tight bounds on the expected absorption time for Ungarian Markov chains on certain lattices, resolving key conjectures.

Findings

Expected absorption time is linear in n for the weak order on S_n.

Expected absorption time has an n^{1-o(1)} lower bound for the Tamari lattice.

Resolved a conjecture by Defant and Li regarding these bounds.

Abstract

In $2023$, Defant and Li defined the Ungarian Markov chain $\mathbf{U}_L$ associated to a finite lattice $L$. This Markov chain has state space $L$, and from any state $x \in L$ transitions to the meet of $\{x\} \cup T$, where $T$ is a randomly selected subset of the elements of $L$ covered by $x$. For any lattice $L$, let $\mathcal{E}(L)$ be the expected number of steps until the maximal element of $L$ transitions into the minimal element in the Ungarian Markov chain. We show that $\mathcal{E}(L)$ is linear in $n$ when $L$ is the weak order on the symmetric group $S_n$, and satisfies an $n^{1-o(1)}$ lower bound when $L$ is the $n^\text{th}$ Tamari lattice. This completely resolves a conjecture by Defant and Li and partially resolves another.