  Giant radio halos are diffuse, Mpc-scale, synchrotron sources located in the
central regions of galaxy clusters and provide the most relevant example of
cluster non-thermal activity. Radio and X-ray surveys allow to investigate the
statistics of halos and may contribute to constrain their origin and evolution.
We investigate the distribution of clusters in the plane X-ray (thermal, L_X)
vs synchrotron (P_{1.4})luminosity, where clusters hosting giant radio halos
trace the P_{1.4}--L_X correlation and clusters without radio halos populate a
region that is well separated from that spanned by the above correlation. The
connection between radio halos and cluster mergers suggests that the cluster
Mpc-scale synchrotron emission is amplified during these mergers and then
suppressed when clusters become more dynamically relaxed. In this context, by
analysing the distribution in the P_{1.4}--L_X plane of clusters from X-ray
selected samples with adequate radio follow up, we constrain the typical
time-scale of evolution of diffuse radio emission in clusters and discuss the
implications for the origin of radio halos. We conclude that cluster
synchrotron emission is suppressed (and amplified) in a time-scale
significantly smaller than 1 Gyr. We show that this constraint appears
difficult to reconcile with the hypothesis that the halo's radio power is
suppressed due to dissipation of magnetic field in galaxy clusters. On the
other hand, in agreement with models where turbulent acceleration plays a role,
present constraints suggest that relativistic electrons are accelerated in
Mpc-scale regions, in connection with cluster mergers and for a time-interval
of about 1 Gyr, and then they cool in a relatively small time-scale, when the
hosting cluster becomes more dynamically relaxed.
