The Variance of the Shock in the Had Process

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 √ t and, as a corollary, prove a central limit theorem.