Plots and simulations

The Long-Range Competition process

The long-range competition (LRC) process is a two-type infection process on the complete graph. Here, the vertices of the complete graph are embedded in the d-dimensional torus (d=1 in the GIFs), and the time for either virus (Type 1 or Type 2) to traverse an edge is an exponential random variable, with a rate that depends on the spatial distance in the torus of the vertices the edge connects. To be more precise, the rate to traverse an edge e is λi |e|^(-αi) for Type i, where λ1,λ2>0 and α1,α2 in [0,d) are model parameters, and |e| denotes the length of the edge e, using a norm on the torus. 

In this paper, Neeladri Maitra and I study the scaling limits of the sizes of the infection types, by the time all vertices are infected by one of the two infections types. α2 λ1 λ2The arguments use a coupling of the two infection processes with independent continuous-time branching processes (CTBPs). Our main theoretical contribution is to control the defect between the LRC process and the CTBPs, even when the defect is rather large. 

In the simulations on the right, we see a competition process where d=1, α1=α2=0.5, and  λ1=λ2=1, so that coexistence occurs. In the first GIF on the right, you can see 4 different plots. The LRC process is on the top-left. On the bottom two plots, you see the CTBPs, where the white vertices correspond to birth in these processes that do not occur in the LRC, i.e. the defects of the coupling. In the top right, you see the size of the infections and the size of the defects of either type.

In the GIF on the bottom, you can see an example with 300 vertices, where the edges in the complete graph along which infections take place are omitted for visibility.