Interface for variants of the contact process

Abstract

We study two one-dimensional variants of the contact process: the contact-and-barrier process, where the population evolves in a region delimited by a randomly moving barrier, and the multitype contact process, in which two species compete for space. The contact-and-barrier process is started with the barrier at the origin and all sites to its right occupied, while the multitype contact process is started from the Heaviside configuration with species 1 to the left of the origin and species 2 to the right. We prove that both models exhibit tight interfaces and that, after centring by an appropriate deterministic speed, the interface position satisfies a central limit theorem. Our analysis relies on a renewal-time method based on a novel construction called patchwork construction, in which the processes are built by concatenating space-time evolutions over successive time intervals of random length, providing a more convenient framework for defining the renewal times that drive the proofs.

Figures (4)

• Figure 1: Schematic representation of the process $(Y_n)_n$, the stopping times $(\kappa_n)_n$, and the random times $(N_k)_k$ from Lemma \ref{['lem_renewal']}. In the graph representing $(\kappa_n)_n$, an arrow from $m$ to $n$ indicates that $\kappa_m=n$.
• Figure 2: Figure \ref{['fig:first']} illustrates both the trail $\mathrm{g}$ and the infection path $\Gamma$ over the graphical construction, while Figure \ref{['fig:second']} shows only the trail $\mathrm{g}\sqcup\Gamma$. We emphasize that a trail itself does not depend on any graphical construction, as it is simply a subset of $\mathbb{Z}\times(-\infty,0]$.
• Figure 3: Illustration of the patchwork elements $T$, $D$, and $\Gamma$. The barrier and its trajectory are shown in orange, the trail $\mathrm{g}$ in blue, and the special infection path $\Gamma$ in purple. The random variable $X$ is shown in purple as well.
• Figure 4: Illustration of the times $\kappa_n$ for a realization of the contact-and-barrier process obtained via the patchwork construction. The barrier and its trajectory are shown in orange; occupied sites are shown in black and empty sites in white. The figure illustrates the connection between Definition \ref{['def_kappa_n']} and Lemma \ref{['lem_renewal']}.

Theorems & Definitions (144)

• Definition 1: Interface for contact-and-barrier process
• Proposition 1
• Theorem 1: Speed of the barrier
• Theorem 2: Tightness of the interface
• Theorem 3: CLT for interface position
• Remark 1
• Definition 2: Interface for multitype contact process
• Theorem 4: Tightness of the interface
• Theorem 5: CLT for interface position
• Definition 3: Infection path