An asymptotic shape theorem for additive random linear growth models

Abstract

In this paper, we define a class of additive random growth models whose growth is at least and at most linear and prove an asymptotic shape theorem for these models. This proof generalizes already known proofs for the classical contact process or some of its variants and allows us to obtain conjectured asymptotic shape theorems for Richardson's model with stirring and the contact process with stirring.

Figures (2)

• Figure 1: Construction of the essential hitting time. At time $u_1(x)$, the site $x$ verifies the property for the first time. At time $v_1(x)$, the process starting from $\Delta_x$ at time $u_1(x)$ does not contain any sites satisfying the property: since it is an absorbing state, the property disappears forever in this process. However, the initial process spreads the property to $x$ at time $u_2(x)$, so the procedure restart at that time. At time $v_2(x)$, the process starting from $\Delta_x$ at time $u_2(x)$ does not contain any sites satisfying the property. Note that instead of pursuing the procedure at the red circle, we start at the first time after the extinction time $v_2(x)$ when $x$ satisfies the property. If we do not proceed as such, then the process obtained does not have the independence properties we want. Finally, the process starting from $\Delta_x$ at time $u_3(x)$ spreads the property forever, therefore the procedure stops.
• Figure 2: The gray block does not contain any bad growth point, so we use the 4 events in the bad growth definition, at arrival times, to control the reinfection time of $x$ from $x_0$.

Theorems & Definitions (28)

• Remark 1
• Definition 1
• Lemma 1
• proof
• Definition 2
• Theorem 2
• Lemma 4
• proof
• Definition 3
• Lemma 5