Picogauss magnetic fields in voids from ultra-high energy cosmic rays
  We argue that ultra-high energy cosmic rays emitted by galaxy clusters result in electric currents in filaments of the large-scale structure that are sufficient to generate magnetic fields in voids of the magnitude of ~1e-12 G and the coherence length of up to tens of Mpc. These fields satisfy both the lower and upper observational bounds on magnetic fields in voids without a need of any primordial component. 
M. S. Pshirkov and P. G. Tinyakov
astro-ph.HE astro-ph.CO