  The statistics of peaks in weak lensing convergence maps is a promising tool
to investigate both the properties of dark matter haloes and constrain the
cosmological parameters. We study how the number of detectable peaks and its
scaling with redshift depend upon the cluster dark matter halo profiles and use
peak statistics to constrain the parameters of the mass - concentration (MC)
relation. We investigate which constraints the Euclid mission can set on the MC
coefficients also taking into account degeneracies with the cosmological
parameters. To this end, we first estimate the number of peaks and its redshift
distribution for different MC relations. We find that the steeper the mass
dependence and the larger the normalisation, the higher is the number of
detectable clusters, with the total number of peaks changing up to $40\%$
depending on the MC relation. We then perform a Fisher matrix forecast of the
errors on the MC relation parameters as well as cosmological parameters. We
find that peak number counts detected by Euclid can determine the normalization
$A_v$, the mass $B_v$ and redshift $C_v$ slopes and intrinsic scatter
$\sigma_v$ of the MC relation to an unprecedented accuracy being
$\sigma(A_v)/A_v = 1\%$, $\sigma(B_v)/B_v = 4\%$, $\sigma(C_v)/C_v = 9\%$,
$\sigma(\sigma_v)/\sigma_v = 1\%$ if all cosmological parameters are assumed to
be known. Should we relax this severe assumption, constraints are degraded, but
remarkably good results can be restored setting only some of the parameters or
combining peak counts with Planck data. This precision can give insight on
competing scenarios of structure formation and evolution and on the role of
baryons in cluster assembling. Alternatively, for a fixed MC relation, future
peaks counts can perform as well as current BAO and SNeIa when combined with
Planck.
