Richardson's model and the contact process with stirring: long time behavior

TL;DR

The paper studies Richardson's model and the contact process with stirring, establishing asymptotic shape theorems for RMS$(\lambda,1)$ and CPS$(\lambda,1,\nu)$ at sufficiently large infection rates via couplings and restart techniques. It proves site fixation for RMS in low dimensions and, more generally, for high enough $\lambda$ in higher dimensions, and it analyzes weak/strong survival on homogeneous trees, identifying two phase transitions with explicit bounds. The authors develop a graphical construction and renewal-coupling framework to compare with RM and CP, derive isoperimetric and growth bounds, and adapt linear-growth theory to obtain the shape results. They also discuss the limitations of extending these results to low infection rates and the challenges posed by stirring with respect to standard block constructions, outlining avenues for future work.

Abstract

We study two famous interacting particle systems, the so-called Richardson's model and the contact process, when we add a stirring dynamics to them. We prove that they both satisfy an asymptotic shape theorem, as their analogues without stirring, but only for high enough infection rates, using couplings and restart techniques. We also show that for Richardson's model with stirring, for high enough infection rates, each site is forever infected after a certain time almost surely. Finally, we study weak and strong survival for both models on a homogeneous infinite tree, and show that there are two phase transitions for certain values of the parameters and the dimension, which is a result similar to what is proved for the contact process.

Figures (6)

• Figure 1: Simulations of the set of once infected sites for some Richardson's models with stirring and contact processes with stirring, after 1 000 000 interactions (infections/healings/exchanges).
• Figure 2: Graphical construction for the CPS on $\mathbb Z$, with initial configuration $A=\{2,4,6\}$, from time $0$ to time $s$. Brown (resp. purple) arrows with I-marks (resp. S-marks) correspond to infection times (resp. stirring time). Green squares correspond to healing times. Open paths starting from an infected site are in red. The set $\xi_s^A$ of infected sites at time $s$, starting from the initial configuration $A$, is the set of intersections between the open paths and the horizontal line of equation $t=s$: here it is equal to $\{1,6\}$. Note that the constraint of following a stirring arrow if and only if the site at the endpoint is healthy is necessary: site $1$ would be healthy at time $s$ if passing by a stirring arrow was mandatory, and site $3$ would be infected at time $s$ if it was optional.
• Figure 3: On the left side, the process $(\eta_t^{\mathrm{CPS}})_{t\ge 0}$, and on the right side, the process $(\xi_t^{\mathrm{CPS}})_{t\ge 0}$. Color conventions are the same as in Figure \ref{['Representation_graphique']}.
• Figure 4: On the left side, the process $(\xi_t^{\mathrm{CPS}})_{t\ge 0}$, and on the right side, the process $(\zeta_t^{\mathrm{CPS}})_{t\ge 0}$. Color conventions are the same as in Figure \ref{['Representation_graphique']}.
• Figure 5: A part of the homogeneous tree $T_2$. On each site, we write its level in the tree.

Theorems & Definitions (15)

• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Proposition 3.4
• Theorem 3.5
• Theorem 3.6
• Lemma 4.1
• proof
• Proposition 4.2
• proof