Birth, Death, and Horizontal Flight: Malthusian flocks with an easy plane in three dimensions

I formulate the theory of three dimensional “Malthusian flocks” – i.e., coherently moving collections of self-propelled entities (such as living creatures) which are being “born” and “dying” during their motion – whose constituents all have a preference for having their velocity vectors lie parallel to the same two-dimensional plane. I determine the universal scaling exponents characterizing such systems exactly, finding that the dynamical exponent $z=3\mathrm{/}2$, the “anisotropy” exponent $\zeta =3\mathrm{/}4$, and the “roughness” exponent $\chi =-1\mathrm{/}2$. I also give the scaling laws implied by these exponents.