Multi-Catalan Tableaux and the Two-Species TASEP
Multi-Catalan Tableaux and the Two-Species TASEP
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多加泰罗尼亚场景和两种 TASEP
DOI：
10.4171/aihpd/30
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Olya Mandelshtam
中科院分区：
文献类型：
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作者：
Olya Mandelshtam
本文的目的是提供一个组合表达式的稳态概率的两个物种PASEP。在这个模型中，有两个种类的粒子，一个“重”和一个“轻”，在一个一维的有限晶格与开放的边界。两个粒子都可以以1和$q$的速率与左右相邻的洞交换位置。此外，当重粒子和轻粒子彼此相邻时，它们可以交换位置，就好像轻粒子是一个洞一样。此外，重粒子可以在晶格的边界处跳跃进出。我们的主要结果是一个组合的解释在$q=0$的平稳分布在某些多加泰罗尼亚tableaux。我们提供了一个明确的行列式公式的稳定状态的概率，以及一些一般的枚举结果，这种情况下。我们还描述了一个马尔可夫过程，这些tableaux项目的两个物种的PASEP，从而直接解释了两者之间的联系。最后，我们给出了一个猜想，扩展我们的公式为平稳分布的情况下，q=1，使用某些两个物种的替代表。
The goal of this paper is to provide a combinatorial expression for the steady state probabilities of the two-species PASEP. In this model, there are two species of particles, one "heavy" and one "light", on a one-dimensional finite lattice with open boundaries. Both particles can swap places with adjacent holes to the right and left at rates 1 and $q$. Moreover, when the heavy and light particles are adjacent to each other, they can swap places as if the light particle were a hole. Additionally, the heavy particle can hop in and out at the boundary of the lattice. Our main result is a combinatorial interpretation for the stationary distribution at $q=0$ in terms of certain multi-Catalan tableaux. We provide an explicit determinantal formula for the steady state probabilities, as well as some general enumerative results for this case. We also describe a Markov process on these tableaux that projects to the two-species PASEP, and thus directly explains the connection between the two. Finally, we give a conjecture that extends our formula for the stationary distribution to the $q=1$ case, using certain two-species alternative tableau.