  Secondary halo properties beyond mass, such as the mass accretion rate (MAR),
concentration, and the half mass scale, are essential in understanding the
formation of large-scale structure and dark matter halos. In this paper, we
study the impact of secondary halo properties on the galaxy-galaxy lensing
observable, $\Delta\Sigma$. We build an emulator trained on N-body simulations
to model $\Delta\Sigma$ and quantify the impact of different secondary
parameters on the $\Delta\Sigma$ profile. We focus on the impact of MAR on
$\Delta\Sigma$. We show that a 3$\sigma$ detection of variations in MAR at
fixed halo mass could be achieved with the Hyper Suprime Cam survey in
combination with a proxy for MAR with scatter
$\sigma_{\Gamma_\mathrm{dyn}|\mathrm{obs}}<1.5$. We show that the full radial
profile of $\Delta\Sigma$ depends on secondary properties at fixed halo mass.
Consequently, an emulator that can perform full shape fitting yields better
than 2 times improvement upon the constraints on MAR than only using the outer
part of the halo. Finally, we highlight that miscentering and MAR impact the
radial profile of $\Delta\Sigma$ in a similar fashion, implying that
miscentering and MAR need to be modeled jointly for unbiased estimates of both
effects. We show that present-day lensing data sets have the statistical
capability to place constraints on halo MAR. Our analysis opens up new
possibilities for observationally measuring the assembly history of the dark
matter halos that host galaxies and clusters.
