The intra-cluster and inter-galactic media (ICM, IGM) that pervade the large scale structure of the Universe are known to be magnetised at sub-micro Gauss to micro Gauss levels and to contain cosmic rays (CRs). The acceleration of CRs and their evolution along with that of magnetic fields in these media is still not well understood. Diffuse radio sources of synchrotron origin associated with the ICM such as radio halos, relics and mini-halos are direct probes of the underlying mechanisms of CR acceleration. Observations with radiotelescopes such as the GMRT, the VLA and the WSRT (0.15 - 2 GHz) have revealed scaling relations between the thermal and non-thermal properties of clusters and favour the role of shocks in the formation of radio relics and of turbulent re-acceleration in the formation of radio halos and mini-halos. Due to the limitations of current radio telescopes, wide-band studies and exploration of low mass and supercluster-scale systems is difficult. The Square Kilometer Array (SKA) is a next generation radio telescope that will operate in the frequency range of 0.05 - 20 GHz with unprecedented sensitivities and resolutions. The expected detection limits of SKA will reveal a few hundred to thousand new radio halos, relics and mini-halos providing the first large and comprehensive samples for their study. The wide frequency coverage along with sensitivity to extended structures will be able to constrain the CR acceleration mechanisms. The higher frequency (> 5 GHz) observations will be able to use the Sunyaev-Zeldovich effect to probe the ICM pressure in addition to the tracers such as lobes of head-tail radio sources. The SKA also opens prospects to detect the off-state radio emission from the ICM predicted by the hadronic models and the turbulent re-acceleration models. [abridged]
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