  Radio and X-ray observations of galaxy clusters probe a direct link between
cluster mergers and giant radio halos (RH), suggesting that these sources can
be used as probes of the cluster merging rate with cosmic time. In this paper
we carry out an explorative study that combines the observed fractions of
merging clusters (fm) and RH (fRH) with the merging rate predicted by
cosmological simulations and attempt to infer constraints on merger properties
of clusters that appear disturbed in X-rays and of clusters with RH. We use
morphological parameters to identify merging systems and analyze the currently
largest sample of clusters with radio and X-ray data (M500>6d14 Msun, and
0.2<z<0.33, from the Planck SZ cluster catalogue). We found that in this sample
fm~62-67% while fRH~44-51%. The comparison of the theoretical f_m with the
observed one allows to constrain the combination (xi_m,tau_m), where xi_m and
tau_m are the minimum merger mass ratio and the timescale of merger-induced
disturbance. Assuming tau_m~ 2-3 Gyr, as constrained by simulations, we find
that the observed f_m matches the theoretical one for xi_m~0.1-0.18. This is
consistent with optical and near-IR observations of clusters in the sample
(xi_m~0.14-0.16). The fact that RH are found only in a fraction of merging
clusters may suggest that merger events generating RH are characterized by
larger mass ratio; this seems supported by optical/near-IR observations of RH
clusters in the sample (xi_min~0.2-0.25). Alternatively, RH may be generated in
all mergers but their lifetime is shorter than \tau_m (by ~ fRH/fm). This is an
explorative study, however it suggests that follow up studies using the
forthcoming radio surveys and adequate numerical simulations have the potential
to derive quantitative constraints on the link between cluster merging rate and
RH at different cosmic epochs and for different cluster masses.
