Tighter Bounds on the Expected Absorbing Time of Ungarian Markov Chains

In 2023, Defant and Li defined the Ungarian Markov chain associated to a lattice L. This Markov chain has state space L, and from any state x ∈ L transitions to the meet of \ ∪ T, where T is a randomly selected subset of the elements of L covered by x. They defined E(L) to be the expected number of steps until the maximal element 1̂ of L transitions into the minimal element 0̂ in the Ungarian Markov chain, and investigated its asymptotics when L is the weak order on Sₙ, or the nth Tamari lattice Tamₙ. This paper resolves a conjecture of Defant and Li and makes partial progress towards another, by proving that E(Sₙ) is linear in n, and proving an n1 - o(1) lower bound on E(Tamₙ).