Emergence of a single cluster in Vicsek's model at very low noise

TL;DR

This paper investigates the Vicsek model at very low noise levels, revealing a phase transition characterized by the emergence of a macroscopic cluster containing a significant fraction of particles.

Contribution

It demonstrates the formation of a macroscopic cluster at low noise and introduces parameters to describe the transition, supported by finite size scaling analysis.

Findings

A bimodal cluster size distribution emerges at low noise.

A phase transition to a macroscopic cluster is identified.

Finite size scaling confirms the transition's critical nature.

Abstract

The classic Vicsek model [Phys.Rev.Lett. {\bf75},1226(1995)] is studied in the regime of very low noise intensities, which is shown to be characterized by a cluster (MC) that contains a macroscopic fraction of the system particles. It is shown that the well-known power-law behavior of the cluster size distribution loses its cutoff becoming bimodal at very low noise intensities: A peak develops for larger sizes to settle the emergence of the MC. The average cluster number m*, is introduced as a parameter that properly describes this change, i.e. a line in the noise-speed phase portrait can be identified to separates both regimes. The average largest cluster parameter also develops large fluctuations at a non zero critical noise. Finite size scaling analysis is performed to show that a phase transition to a macroscopic cluster is taking place. Consistency of the results with the literature is also checked and commented upon.