A determinantal formula for Catalan tableaux and TASEP probabilities
A determinantal formula for Catalan tableaux and TASEP probabilities
复制标题
Catalan tableaux 和 TASEP 概率的行列式
DOI：
10.1016/j.jcta.2014.12.005
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Olya Mandelshtam
中科院分区：
文献类型：
--
作者：
Olya Mandelshtam
本文给出了具有开放边界的完全不对称简单不相容过程（TASEP）各态稳态概率的行列式，TASEP是一个被广泛研究的具有丰富组合结构的一维粒子模型。这些稳态概率是通过列举加泰罗尼亚表来计算的，这些表是由满足行和列上的某些条件的充满α和β的杨氏图。我们构造了一个从加泰罗尼亚表到Young图上加权点阵路径的双射，并在Narayana计算Young图上未加权点阵路径的公式的基础上，用行列式公式列举了这些路径。最后，我们给出了在行上满足给定条件的加泰罗尼亚表的枚举公式，该公式对应于具有n个位点的晶格上的TASEP中恰好有k个位点被粒子占据的稳态概率。这个公式是Narayana数的α/β推广。
We present a determinantal formula for the steady state probability of each state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open boundaries, a 1D particle model that has been studied extensively and displays rich combinatorial structure. These steady state probabilities are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with α's and β's that satisfy some conditions on the rows and columns. We construct a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram, and from this we enumerate the paths with a determinantal formula, building upon a formula of Narayana that counts unweighted lattice paths on a Young diagram. Finally, we provide a formula for the enumeration of Catalan tableaux that satisfy a given condition on the rows, which corresponds to the steady state probability that in the TASEP on a lattice with n sites, precisely k of the sites are occupied by particles. This formula is an α/β generalization of the Narayana numbers.