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, Volume 35, Issue 1, pp 93–109 | Cite as

Markov Chains on Graded Posets

Compatibility of Up-Directed and Down-Directed Transition Probabilities
  • Kimmo Eriksson
  • Markus Jonsson
  • Jonas Sjöstrand
Open Access
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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 up chain or down chain). The second type toggles between two adjacent rank levels (an up-and-down chain). We introduce two compatibility concepts between the up-directed transition probabilities (an up rule) and the down-directed (a 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.

Keywords

Graded poset Markov chain Young diagram Young’s lattice Limit shape 

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Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of Education, Culture and CommunicationMälardalen UniversityVästeråsSweden
  2. 2.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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