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Asymptotics of two-point correlations in the multi-species q-TAZRP

Jeffrey Kuan, Zhengye Zhou

Abstract

A previous paper by the authors found explicit contour integral formulas for certain joint moments of the multi-species q-TAZRP (totally asymmetric zero range process), using algebraic methods. These contour integral formulas have a "pseudo-factorized" form which makes asymptotic analysis simpler. In this brief note, we use those contour integral formulas to find the asymptotics of the two-point correlations. As expected, the term arising from the "shift-invariance" makes a non-trivial asymptotic contribution.

Asymptotics of two-point correlations in the multi-species q-TAZRP

Abstract

A previous paper by the authors found explicit contour integral formulas for certain joint moments of the multi-species q-TAZRP (totally asymmetric zero range process), using algebraic methods. These contour integral formulas have a "pseudo-factorized" form which makes asymptotic analysis simpler. In this brief note, we use those contour integral formulas to find the asymptotics of the two-point correlations. As expected, the term arising from the "shift-invariance" makes a non-trivial asymptotic contribution.
Paper Structure (7 sections, 2 theorems, 29 equations, 1 figure)

This paper contains 7 sections, 2 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Result
  3. Definition of Model
  4. Main Theorem
  5. Proof
  6. Asymptotics of $q_1,q_2,q_3$
  7. Asymptotics of $q_4,q_5,q_6$

Key Result

Theorem 1

Suppose $\boldsymbol\xi$ has initial conditions with $n_1$ species 0 particles at site $x_1$ and $n_2$ species 1 particles at site $x_2$, or in other words and $\boldsymbol\xi$ evolves as a $q$--TAZRP with total asymmetry to the right. Let $y_1$ and $y_2$ be some lattice sites, and additionally assuming that $x_1<x_2$. The dual process $\boldsymbol\eta$ has 1 species 0 particles at site $y_1$, 1

Figures (1)

  • Figure 1: The jump rates for the $q$--TAZRP.

Theorems & Definitions (4)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 4