Non-equilibrium Fluctuations of the Weakly Asymmetric Normalized Binary Contact Path Process

This paper is a further investigation of the problem studied in cite{xue2020hydrodynamics}, where the authors proved a law of large numbers for the empirical measure of the weakly asymmetric normalized binary contact path process on $mathbb{Z}^d,, d geq 3$, and then conjectured that a central limit theorem should hold under a non-equilibrium initial condition. We prove that the aforesaid conjecture is true when the dimension $d$ of the underlying lattice and the infection rate $lambda$ of the process are sufficiently large.