The super-critical contact process has a spectral gap

Abstract

We consider the super-critical contact process on $\mathbb{Z}^d$. It is known that measures which dominate the upper invariant measure $\mu$ converge exponentially fast to $\mu$. However, the same is not true for measures which are below $\mu$, as the time to infect a large empty region is related to its diameter. The result of this paper is the existence of a spectral gap in $L^2(\mu)$, that is, the spectrum of the generator is empty inside an open strip $\{z\in\mathbb{C}: -\lambda<\Im(z)<0\}$ of the complex plane. This is equivalent to the fact that the variance of the semi-group of the contact process decays exponentially fast. It is perhaps surprising that the existence of the spectral gap has not been proven before. One of the reasons is that the contact process is non-reversible, and hence many methods from spectral theory are not applicable.