Shock fluctuations for the Hammersley process

TL;DR

This paper analyzes the fluctuations of shocks in the Hammersley process, establishing a central limit theorem and variance estimates for the shock position and extending results to first-class particles.

Contribution

It provides a new CLT and variance formula for shocks in the Hammersley process, including first-class particles, using a novel proof method.

Findings

Central limit theorem for shock position

Variance of shock grows linearly with time

Results extend to first-class particles

Abstract

We consider the Hammersley interacting particle system starting from a shock initial profile with densities $\lambda,\rho\in\mathbb{R}$ ($\lambda > \rho$). The microscopic shock is taken as the position of a second-class particle initially at the origin, and the main results are: a central limit theorem for the shock; the variance of the shock equals $2[\lambda\rho(\lambda - \rho)]^{-1}t + O(t^{2/3})$. By using the same method of proof, we also prove similar results for first-class particles.