  We study the role of noise on the nature of the transition to collective
motion in dry active matter. Starting from field theories that predict a
continuous transition at the deterministic level, we show that fluctuations
induce a density-dependent shift of the onset of order, which in turns changes
the nature of the transition into a phase-separation scenario. Our results
apply to a range of systems, including the topological models in which
particles interact with a fixed number of nearest neighbors, which were
believed so far to exhibit a continuous onset of order. Our analytical
predictions are confirmed by numerical simulations of fluctuating hydrodynamics
and microscopic models.
