  The stability properties of a low density ultra relativistic pair beam
produced in the intergalactic medium by multi-TeV gamma-ray photons from
blazars are analyzed. The problem is relevant for probes of magnetic field in
cosmic voids through gamma-ray observations. In addition, dissipation of such
beams could affect considerably the thermal history of the intergalactic medium
and structure formation. We use a Monte Carlo method to quantify the properties
of the blazar induced electromagnetic shower, in particular the bulk Lorentz
factor and the angular spread of the pair beam generated by the shower, as a
function of distance from the blazar itself. We then use linear and nonlinear
kinetic theory to study the stability of the pair beam against the growth of
electrostatic plasma waves, employing the Monte Carlo results for our
quantitative estimates. We find that the fastest growing mode, like any
perturbation mode with even a very modest component perpendicular to the beam
direction, cannot be described in the reactive regime. Due to the effect of
non-linear Landau damping, which suppresses the growth of plasma oscillations,
the beam relaxation timescale is found significantly longer than the inverse
Compton loss time. Finally, density inhomogeneities associated with cosmic
structure induce loss of resonance between the beam particles and plasma
oscillations, strongly inhibiting their growth. We conclude that relativistic
pair beams produced by blazars in the intergalactic medium are stable on
timescales long compared to the electromagnetic cascade's. There appears to be
little or no effect of pair-beams on the intergalactic medium.
