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.
Toric tableaux and the inhomogeneous two-species TASEP on a ring
环上的非均匀两种粒子TASEP是一个排它过程，它描述了不同种类的粒子在环上顺时针跳跃，参数给出了不同种类的跳跃率。我们引入一个组合对象，我们称之为\emph{toric rhombic alternative tableaux}，这是某些填充的tableaux上的三角形格子平铺菱形，并在双射与法拉利和马丁的研究\emph{multiline queues}。使用表格，我们得到一个公式的平稳概率，这TASEP，专门的Ayyer和Linusson的结果。此外，我们获得了一个明确的行列式公式，这些概率，并定义了一个马尔可夫链的tableaux项目的两个物种TASEP环上。
Jeu de taquin dynamics on infinite Young tableaux and second class particles
我们研究了一个无限版本的“jeu de taquin”滑动游戏，它可以被认为是一个自然的保测变换的集合上的无限杨tableaux配备Plancherel概率测度。我们使用表示论的方法表明，Robinson-Schensted-Knuth（$\operatorname {RSK}$）算法给出了一个同构之间的测度保持动力系统和单侧移位动力学的序列上的独立和同分布的随机变量均匀分布的单位区间。我们还证明了由变换诱导的jeu de taquin轨迹是从原点沿随机方向发出的渐近直线，其分布被显式计算，并表明这一结果可以解释为Plancherel-TASEP粒子系统中第二类粒子的极限速度的陈述（与Plancherel增长相关的完全不对称简单排它过程的变体），与普通TASEP中第二类粒子的早期结果类似。
