Quantitative hydrodynamics for a generalized contact model

TL;DR

This paper establishes a precise hydrodynamic limit for an interacting particle system inspired by neuron models, showing optimal convergence rates and Gaussian fluctuations governed by a stochastic linear equation.

Contribution

It provides the first quantitative hydrodynamic limit with optimal convergence rate and characterizes the Gaussian fluctuations for a generalized contact process.

Findings

Optimal $L^2$-speed of convergence $oxed{ ext{O}(n^{d/2})}$.

Fluctuations are Gaussian and described by an inhomogeneous stochastic linear equation.

Results apply to a $d$-dimensional torus of size $n$.

Abstract

We derive a quantitative version of the hydrodynamic limit 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 $\mathcal O(n^{d/2})$. In addition, we show that the typical fluctuations around the aforementioned hydrodynamic limit are Gaussian, and governed by a inhomogeneous stochastic linear equation.