Markov chains on graded posets: Compatibility of up-directed and down-directed transition probabilities

TL;DR

This paper studies Markov chains on graded posets, introducing compatibility concepts between up and down transition probabilities, and applies these to prove a limit shape conjecture and analyze chain reversibility.

Contribution

It introduces compatibility concepts for transition probabilities on graded posets and applies them to prove a limit shape conjecture and analyze reversibility of chains.

Findings

Proved a conjecture about a limit shape for a process on Young's lattice.

Established conditions for reversibility of up and down chains.

Analyzed existence of chains when the rank function is unbounded.

Abstract

We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an \emph{up chain} or \emph{down chain}). The second type toggles between two adjacent rank levels (an \emph{up-and-down chain}). We introduce two compatibility concepts between the up-directed transition probabilities (an \emph{up rule}) and the down-directed (a \emph{down rule}), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young's lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.