The variance of the shock in the HAD process

TL;DR

This paper analyzes the variance of shock positions in the HAD process with sinks and sources, deriving explicit formulas for mean and variance, and establishing a central limit theorem for the shock position's fluctuations.

Contribution

It provides explicit formulas for the mean and variance of shock positions in the HAD process and characterizes their dependence on initial conditions.

Findings

Mean and variance of shock position are linear in time.

Explicit constants depend on boundary densities.

Central limit theorem for shock fluctuations.

Abstract

We consider the Hammersley-Aldous-Diaconis (HAD) process with sinks and sources such that there is a microscopic shock at every time $t$; denote $Z(t)$ its position. We show that the mean and variance of $Z(t)$ are linear functions of $t$ and compute explicitely the respective constants in function of the left and right densities. Furthermore, we describe the dependence of $Z(t)$ on the initial configuration in the scale $\sqrt t$ and, as a corollary, prove a central limit theorem.