Self-propelled Vicsek particles at low speed and low density

We study through numerical simulation the Vicsek model for very low speeds and densities. We consider scalar noise in 2-d and 3-d, and vector noise in 3-d. We focus on the behavior of the critical noise with density and speed, trying to clarify seemingly contradictory earlier results. We find that, for scalar noise, the critical noise is a power law both in density and speed, but although we confirm the density exponent in 2-d, we find a speed exponent different from earlier reports (we consider lower speeds than previous studies). On the other hand, for the vector noise case we find that the dependence of the critical noise cannot be separated as a product of power laws in speed and density. Finally, we study the dependence of the relaxation time with speed and find the same power law in 2-d and 3-d, with and exponent that depends on whether the noise is above or below the critical value.