Q-zero range has random walking shocks

TL;DR

This paper investigates the existence of random walking shock measures in 1D asymmetric zero range processes, identifying the q-zero range process as a unique new example and exploring shock interactions.

Contribution

It demonstrates that the q-zero range process is the only new model with random walking shocks among nearest neighbor 1D asymmetric zero range processes, expanding understanding of shock phenomena.

Findings

q-zero range process is the only new example with random walking shocks

Interaction of shocks is more complex in q-zero range processes

Confirmed known models and identified the unique new case

Abstract

... but no other surprises in the zero range world. We check all nearest neighbour 1-dimensional asymmetric zero range processes for random walking product shock measures as demonstrated already for a few cases in the literature. We find the totally asymmetric version of the celebrated q-zero range process as the only new example besides an already known model of doubly infinite occupation numbers and exponentially increasing jump rates. We also examine the interaction of shocks, which appears somewhat more involved for q-zero range than in the already known cases.