Rowmotion Markov chains

Colin Defant, Rupert Li, Evita Nestoridi

Abstract: Rowmotion is a certain well-studied bijective operator on the distributive lattice J(P)<math><mi is="true">J</mi><mo stretchy="false" is="true">(</mo><mi is="true">P</mi><mo stretchy="false" is="true">)</mo></math> of order ideals of a finite poset P. We introduce the rowmotion Markov chain MJ(P)<math><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">M</mi></mrow><mrow is="true"><mi is="true">J</mi><mo stretchy="false" is="true">(</mo><mi is="true">P</mi><mo stretchy="false" is="true">)</mo></mrow></msub></math> by assigning a probability px<math><msub is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">x</mi></mrow></msub></math> to each x∈P<math><mi is="true">x</mi><mo is="true">∈</mo><mi is="true">P</mi></math> and using these probabilities to insert randomness into the original definition of rowmotion. More generally, we introduce a very broad family of toggle Markov chains inspired by Striker's notion of generalized toggling. We characterize when toggle Markov chains are irreducible, and we show that each toggle Markov chain has a remarkably simple stationary distribution.We also provide a second generalization of rowmotion Markov chains to the context of semidistrim lattices. Given a semidistrim lattice L, we assign a probability pj<math><msub is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub></math> to each join-irreducible element j of L and use these probabilities to construct a rowmotion Markov chain ML<math><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">M</mi></mrow><mrow is="true"><mi is="true">L</mi></mrow></msub></math>. Under the assumption that each probability pj<math><msub is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub></math> is strictly between 0 and 1, we prove that ML<math><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">M</mi></mrow><mrow is="true"><mi is="true">L</mi></mrow></msub></math> is irreducible. We also compute the stationary distribution of the rowmotion Markov chain of a lattice obtained by adding a minimal element and a maximal element to a disjoint union of two chains.We bound the mixing time of ML<math><msub is="true"><mrow is="true"><mi mathvariant="bold" is="true">M</mi></mrow><mrow is="true"><mi is="true">L</mi></mrow></msub></math> for an arbitrary semidistrim lattice L. In the special case when L is a Boolean lattice, we use spectral methods to obtain much stronger estimates on the mixing time, showing that rowmotion Markov chains of Boolean lattices exhibit the cutoff phenomenon.