Parameter Permutation Symmetry in Particle Systems and Random Polymers
Parameter Permutation Symmetry in Particle Systems and Random Polymers
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DOI：
10.3842/sigma.2021.021
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发表时间：
2019-12
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影响因子：
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通讯作者：
L. Petrov
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文献类型：
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作者：
L. Petrov
当我们为每个粒子$x_i$配备自己的跳速参数$\nu_i$时，一维空间中的许多可积随机粒子系统（如TASEP -完全不对称简单排斥过程-及其各种变形，ASEP是一个明显的例外）仍然是可积的。从阶跃初始构型出发的系统中每个粒子$x_n(t)$的分布以对称的方式依赖于参数$\nu_j$, $j\le n$，这是可积性的结果。因此，参数的移位$\nu_n \leftrightarrow \nu_{n+1}$只影响$x_n(t)$的分布。对于$q$ -Hahn TASEP及其退化（即$q$ -TASEP和β聚合物），我们将转置$\nu_n \leftrightarrow \nu_{n+1}$实现为作用于单个粒子$x_n(t)$的显式马尔可夫交换算子。对于β聚合物，交换算子可以解释为考虑聚合物的晶格的简单修饰。我们的主要工具是马尔可夫对偶性和关节矩的轮廓积分公式。特别是，我们的构造导致了一个连续时间马尔可夫过程$\mathsf{Q}^{(\mathsf{t})}$，它保留了$q$ -TASEP的时间$\mathsf{t}$分布（具有步进初始配置，其中$\mathsf{t}\in \mathbb{R}_{>0}$是固定的）。对偶系统是随机$q$ -玻色子系统的某种暂态修正。我们用$q$ -TASEP的$q$ -矩确定了该瞬态过程的渐近生存概率，并用它来表示具有任意初始数据的过程$\mathsf{Q}^{(\mathsf{t})}$收敛于其平稳分布。设置$q=0$，我们恢复了最近在arXiv:1907.09155[数学]中建立的通常TASEP的结果。基于二维交错粒子的吉布斯系综的另一种方法。
Many integrable stochastic particle systems in one space dimension (such as TASEP - Totally Asymmetric Simple Exclusion Process - and its various deformations, with a notable exception of ASEP) remain integrable when we equip each particle $x_i$ with its own jump rate parameter $\nu_i$. It is a consequence of integrability that the distribution of each particle $x_n(t)$ in a system started from the step initial configuration depends on the parameters $\nu_j$, $j\le n$, in a symmetric way. A transposition $\nu_n \leftrightarrow \nu_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$. For $q$-Hahn TASEP and its degenerations (namely, $q$-TASEP and beta polymer) we realize the transposition $\nu_n \leftrightarrow \nu_{n+1}$ as an explicit Markov swap operator acting on the single particle $x_n(t)$. For beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are Markov duality and contour integral formulas for joint moments. In particular, our constructions lead to a continuous time Markov process $\mathsf{Q}^{(\mathsf{t})}$ preserving the time $\mathsf{t}$ distribution of the $q$-TASEP (with step initial configuration, where $\mathsf{t}\in \mathbb{R}_{>0}$ is fixed). The dual system is a certain transient modification of the stochastic $q$-Boson system. We identify asymptotic survival probabilities of this transient process with $q$-moments of the $q$-TASEP, and use this to show convergence of the process $\mathsf{Q}^{(\mathsf{t})}$ with arbitrary initial data to its stationary distribution. Setting $q=0$, we recover the results about the usual TASEP established recently in arXiv:1907.09155 [math.PR] by a different approach based on Gibbs ensembles of interlacing particles in two dimensions.