  The interaction of TeV photons from blazars with the extragalactic background
light produces a relativistic beam of electron-positron pairs streaming through
the intergalactic medium (IGM). The fate of the beam energy is uncertain. By
means of two- and three-dimensional particle-in-cell simulations, we study the
non-linear evolution of dilute ultra-relativistic pair beams propagating
through the IGM. We explore a wide range of beam Lorentz factors gamma_b>>1 and
beam-to-plasma density ratios alpha<<1, so that our results can be extrapolated
to the extreme parameters of blazar-induced beams (gamma_b~10^6 and
alpha~10^(-15), for the most powerful blazars). For cold beams, we show that
the oblique instability governs the early stages of evolution, but its
exponential growth terminates - due to self-heating of the beam in the
transverse direction - when only a negligible fraction
~(alpha/gamma_b)^(1/3)~10^(-7) of the beam energy has been transferred to the
IGM plasma. Further relaxation of the beam proceeds through quasi-longitudinal
modes, until the momentum dispersion in the direction of propagation saturates
at DeltaP_{b,parallel}/gamma_b m_e c ~ 0.2. This corresponds to a fraction ~10%
of the beam energy being ultimately transferred to the IGM plasma, irrespective
of gamma_b or alpha. If the initial dispersion in beam momentum satisfies
DeltaP_{b,parallel}/gamma_b m_e c > 0.2 (as typically expected for
blazar-induced beams), the fraction of beam energy deposited into the IGM is
much smaller than ~10%. It follows that at least ~90% of the beam energy is
still available to power the GeV emission produced by inverse Compton
up-scattering of the Cosmic Microwave Background by the beam pairs.
