  The combination of galaxy-galaxy lensing (GGL) with galaxy clustering is one
of the most promising routes to determining the amplitude of matter clustering
at low redshifts. We show that extending clustering+GGL analyses from the
linear regime down to $\sim 0.5 \, h^{-1}$ Mpc scales increases their
constraining power considerably, even after marginalizing over a flexible model
of non-linear galaxy bias. Using a grid of cosmological N-body simulations, we
construct a Taylor-expansion emulator that predicts the galaxy autocorrelation
$\xi_{\text{gg}}(r)$ and galaxy-matter cross-correlation $\xi_{\text{gm}}(r)$
as a function of $\sigma_8$, $\Omega_m$, and halo occupation distribution (HOD)
parameters, which are allowed to vary with large scale environment to represent
possible effects of galaxy assembly bias. We present forecasts for a fiducial
case that corresponds to BOSS LOWZ galaxy clustering and SDSS-depth weak
lensing (effective source density $\sim 0.3$ arcmin$^{-2}$). Using tangential
shear and projected correlation function measurements over $0.5 \leq r_p \leq
30 \, h^{-1}$ Mpc yields a 1.8% constraint on the parameter combination
$\sigma_8\Omega_m^{0.58}$, a factor of two better than a constraint that
excludes non-linear scales ($r_p > 2 \, h^{-1}$ Mpc, $4 \, h^{-1}$ Mpc for
$\gamma_t,w_p$). Much of this improvement comes from the non-linear clustering
information, which breaks degeneracies among HOD parameters that would
otherwise degrade the inference of matter clustering from GGL. Increasing the
effective source density to $3$ arcmin$^{-2}$ sharpens the constraint on
$\sigma_8\Omega_m^{0.58}$ by a further factor of two. With robust modeling into
the non-linear regime, low-redshift measurements of matter clustering at the
1-percent level with clustering+GGL alone are well within reach of current data
sets such as those provided by the Dark Energy Survey.
