The structure and assembly history of cluster-size haloes in
  Self-Interacting Dark Matter
  We perform dark-matter-only simulations of 28 relaxed massive cluster-sized
haloes for Cold Dark Matter (CDM) and Self-Interacting Dark Matter (SIDM)
models, to study structural differences between the models at large radii,
where the impact of baryonic physics is expected to be very limited. We find
that the distributions for the radial profiles of the density, ellipsoidal axis
ratios, and velocity anisotropies ($\beta$) of the haloes differ considerably
between the models (at the $\sim1\sigma$ level), even at $\gtrsim10\%$ of the
virial radius, if the self-scattering cross section is $\sigma/m_\chi=1$ cm$^2$
gr$^{-1}$. Direct comparison with observationally inferred density profiles
disfavours SIDM for $\sigma/m_\chi=1$ cm$^2$ gr$^{-1}$, but in an intermediate
radial range ($\sim3\%$ of the virial radius), where the impact of baryonic
physics is uncertain. At this level of the cross section, we find a narrower
$\beta$ distribution in SIDM, clearly skewed towards isotropic orbits, with no
SIDM (90\% of CDM) haloes having $\beta>0.12$ at $7\%$ of the virial radius. We
estimate that with an observational sample of $\sim30$ ($\sim10^{15}$
M$_\odot$) relaxed clusters, $\beta$ can potentially be used to put competitive
constraints on SIDM, once observational uncertainties improve by a factor of a
few. We study the suppression of the memory of halo assembly history in SIDM
clusters. For $\sigma/m_\chi=1$ cm$^2$ gr$^{-1}$, we find that this happens
only in the central halo regions ($\sim1/4$ of the scale radius of the halo),
and only for haloes that assembled their mass within this region earlier than a
formation redshift $z_f\sim2$. Otherwise, the memory of assembly remains and is
reflected in ways similar to CDM, albeit with weaker trends.
