Lanes and lattice structures in a repulsive model for self-propelled agents

We investigate a simple Vicsek-type rule-based model for self-propelled particles, where each particle orients itself antiparallel to the average orientation of particles within a defined neighborhood of radius $R$. The particle orientation is updated asynchronously and randomly across the system. In steady state, particles self-organize into clusters-despite the repulsive interaction-and form two interwoven hexagonal lattices moving in opposite directions chosen spontaneously. Increasing noise in the reorientation step reduces the laning effect, but the global crystalline order remains intact at sufficiently high densities. The mean-squared displacement exhibits super-diffusive growth $\sim t^{3/2}$ in the transient phase, transitioning to ballistic motion $\sim t^2$ in the steady state in the high density and zero noise regime. With an increase in noise and/or decrease in density, the mean-squared displacement grows diffusively $\sim t$. We observe a cutoff for the ratio $\frac{R}{L} \sim 0.2-0.3$, below which laning and crystallization is achieved, suggesting a local but non-microscopic sphere of influence is required to initiate laning.