Synchronization of Vicsek Model with Large Population

The Vicsek model can be used to describe a basic class of multi-agent systems with local interactions: each agent has the tendency to behave as other agents do in its neighborhood. Through computer simulations, Vicsek et al. (1995) showed that such simple local interactions may lead to certain kind of cooperative phenomenon (synchronization) of the overall system, if the size of the system population is large. Since this model is of fundamental importance in understanding multi-agent systems, it has attracted much attention from researchers in recent years. In this paper, we will present a comprehensive theoretical analysis of the Vicsek model in a random framework with large population. To be precise, we will show that if both the interaction radius r and the agents´ moving velocity v decrease as the population size n increases, but satisfy a certain constraint on the decreasing rates, then the overall system will synchronize for large n. The proofs are based on the recent work of Tang and Guo, and involves the use of spectral graph theory and double array Martingale estimation theory.