Rowmotion 马尔可夫链
Rowmotion Markov Chains
AI总结：
本文将确定性 rowmotion 推广为带概率参数的切换与半分配格马尔可夫链，刻画不可约性、平稳分布及混合时间，并证明布尔格情形具有 cutoff 现象。
摘要：
Rowmotion 是有限偏序集 $P$ 的序理想构成的分配格 $J(P)$ 上一个被充分研究的双射算子。我们通过为每个 $x\in P$ 指定概率 $p_x$，并利用这些概率将随机性引入 rowmotion 的原始定义，引入了 rowmotion 马尔可夫链 ${\bf M}_{J(P)}$。更一般地，我们受 Striker 的广义切换概念启发，引入了一个非常广泛的切换马尔可夫链族。我们刻画了切换马尔可夫链何时不可约，并证明每个切换马尔可夫链都具有异常简单的平稳分布。我们还给出了 rowmotion 马尔可夫链在半分配格语境下的第二种推广。给定半分配格 $L$，我们为 $L$ 的每个并不可约元素 $j$ 指定概率 $p_j$，并利用这些概率构造 rowmotion 马尔可夫链 ${\bf M}_L$。在每个概率 $p_j$ 严格介于 0 和 1 之间的假设下，我们证明 ${\bf M}_{L}$ 是不可约的。我们还计算了在两条链的不交并上添加一个极小元和一个极大元所得格的 rowmotion 马尔可夫链的平稳分布。对于任意半分配格 $L$，我们界定了 ${\bf M}_{L}$ 的混合时间。在 $L$ 为布尔格的特殊情形下，我们使用谱方法获得了强得多的混合时间估计，表明布尔格的 rowmotion 马尔可夫链展现出截断现象。
英文摘要：
Rowmotion is a certain well-studied bijective operator on the distributive lattice $J(P)$ of order ideals of a finite poset $P$. We introduce the rowmotion Markov chain ${\bf M}_{J(P)}$ by assigning a probability $p_x$ to each $x\in P$ 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 $p_j$ to each join-irreducible element $j$ of $L$ and use these probabilities to construct a rowmotion Markov chain ${\bf M}_L$. Under the assumption that each probability $p_j$ is strictly between $0$ and $1$, we prove that ${\bf M}_{L}$ 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 ${\bf M}_{L}$ 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.
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