Finite-size scaling analysis and dynamic study of the critical behavior of a model for the collective displacement of self-driven individuals

The Vicsek model (VM) [T. Vicsek et al., Phys. Rev. Lett. 75, 1226 (1995)], for the description of the collective behavior of self-driven individuals, assumes that each of them adopts the average direction of movement of its neighbors, perturbed by an external noise. A second-order transition between a state of ordered collective displacement (low-noise limit) and a disordered regime (high-noise limit) was found early on. However, this scenario has recently been challenged by Gregory and Chate [G. Gregory and H. Chate, Phys. Rev. Lett. 92, 025702 (2004)] who claim that the transition of the VM may be of first order. By performing extensive simulations of the VM, which are analyzed by means of both finite-size scaling methods and a dynamic scaling approach, we unambiguously demonstrate the critical nature of the transition. Furthermore, the complete set of critical exponents of the VM, in d=2 dimensions, is determined. By means of independent methods - i.e., stationary and dynamic measurements - we provide two tests showing that the standard hyper-scaling relationship dv-2 beta = gamma holds, where beta, nu, and gamma are the order parameter, correlation length, and "susceptibility" critical exponents, respectively. Furthermore, we established that at criticality, the correlation length grows according to xi similar to t(1/z), with z similar or equal to 1.27(3), independently of the degree of order of the initial configuration, in marked contrast with the behavior of the XY model.