  Recent improvements in the capabilities of low frequency radio telescopes
provide a unique opportunity to study thermal and non-thermal properties of the
cosmic web. We argue that the diffuse, polarized emission from giant radio
relics traces structure formation shock waves and illuminates the large-scale
magnetic field. To show this, we model the population of shock-accelerated
relativistic electrons in high-resolution cosmological simulations of galaxy
clusters and calculate the resulting radio synchrotron emission. We find that
individual shock waves correspond to localized peaks in the radio surface
brightness map which enables us to measure Mach numbers for these shocks. We
show that the luminosities and number counts of the relics strongly depend on
the magnetic field properties, the cluster mass and dynamical state. By
suitably combining different cluster data, including Faraday rotation measures,
we are able to constrain some macroscopic parameters of the plasma at the
structure formation shocks, such as models of turbulence. We also predict upper
limits for the properties of the warm-hot intergalactic medium, such as its
temperature and density. We predict that the current generation of radio
telescopes (LOFAR, GMRT, MWA, LWA) have the potential to discover a
substantially larger sample of radio relics, with multiple relics expected for
each violently merging cluster. Future experiments (SKA) should enable us to
further probe the macroscopic parameters of plasma physics in clusters.
