  Charged and quasi-neutral beams propagating through an unmagnetised plasma
are subject to numerous collisionless instabilities on the small scale of the
plasma skin depth. The electrostatic two-stream instability, driven by
longitudinal and transverse wakefields, dominates for dilute beams. This leads
to modulation of the beam along the propagation direction and, for wide beams,
transverse filamentation. A three-dimensional spatiotemporal two-stream theory
for warm beams with a finite extent is developed. Unlike the cold beam limit,
diffusion due to a finite emittance gives rise to a dominant wavenumber, and a
cut-off wavenumber above which filamentation is suppressed. Particle-in-cell
simulations give excellent agreement with the theoretical model. This work
provides deeper insights into the effect of diffusion on filamentation of
finite beams, crucial for comprehending plasma-based accelerators in laboratory
and cosmic settings.
