r_{\rm vir}/r_{\rm s}$, where $r_{\rm vir}$ is the virial radius of the halo and $r_{\rm s}$ is the inner “transitional” radius appearing in the @NFW:1996 profile, at which the density profile changes slope. It is often alternatively defined via the rotation curve of the halo, as the ratio between $V_{\rm max}$ and $V_{\rm vir}$, where $V_{\rm max}$ is the peak value of the rotation curve, $V^2_{\rm c} = G\ M(r)/r$, and $V_{\rm vir}$ the virial velocity of the halo, $V_{\rm vir} \equiv V_c(r_{\rm vir})$ [@Bullock:2001; @Gao:2007]. We utilize the latter definition here, which is directly calculable from simulation data and does not require any model fitting.

In order to explore the variation in clustering and halo occupation with halo age and concentration, following Z18, we rank the haloes by these properties and identify the 20 per cent oldest and youngest haloes (based on their formation time) and (separately) the 20 per cent of haloes that are most or least concentrated. These divisions are made in 0.1 dex bins of halo mass, so as to factor out the influence of the changing halo mass function on these properties; the 20 per cent extremes of the distribution set up in this way 