Quantitative hydrodynamics for a generalized contact model

We derive a quantitative version of the hydrodynamic limit obtained in Chariker et al. (2023) for an interacting particle system inspired by integrate-and-fire neuron models. More precisely, we show that the $L^2$-speed of convergence of the empirical density of states in a generalized contact process defined over a $d$-dimensional torus of size $n$ is of the optimal order $O\left(n^{d/2}\right)$. In addition, we show that the typical fluctuations around the aforementioned hydrodynamic limit are Gaussian, and governed by an inhomogeneous stochastic linear equation.