Correlations in the multispecies PASEP on a ring

TL;DR

This work extends exact correlation analysis from the multispecies TASEP to the partially asymmetric case on a finite ring by leveraging Martin's multiline process and the $q$-bully path algorithm. It derives explicit closed-form expressions for the two-point stationary correlations $c_{i,j}^q(n)$ in terms of the baseline $q=0$ results and $q$-integers $[k]_q$, distinguishing the cases $i>j$ and $i<j$. The methodology relies on weighted projections of linked multiline queues and a projection principle that enables lumping multispecies dynamics to lower-species subsystems. The results unify TASEP and PASEP correlations in a combinatorial framework and provide exact formulas for adjacent-site correlations in multispecies rings with potential connections to Macdonald-type structures through the multiline queue construction.

Abstract

Ayyer and Linusson studied correlations in the multispecies TASEP on a ring (Trans AMS, 2017) using a combinatorial analysis of the multiline queues construction defined by Ferrari and Martin (AOP, 2008). It is natural to explore whether an analogous application of appropriate multiline queues could give similar results for the partially asymmetric case. In this paper, we solve this problem of correlations of adjacent particles on the first two sites in the multispecies PASEP on a finite ring. We use the multiline processes defined by Martin (EJP, 2020), the dynamics of which also depend on the asymmetry parameter $q$, to compute the correlations.

Figures (4)

• Figure 1: A multiline queue of type $(2,1,2,2,6)$
• Figure 2: A linked multiline queue of type $(2,1,2,2,6)$
• Figure 4: Linked multiline queues of an MLQ of type (3,2,3).
• Figure 5: An $L$MLQ of type $(2,1,2,2,6)$ with weight $\frac{q^4}{[6]_q![5]_q}$

Theorems & Definitions (13)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Example 2.4
• Theorem 3.1
• Remark 3.2
• Example 3.3
• Lemma 3.4
• proof
• proof : Proof of Theorem \ref{['th:etaPASEP']}