  Unprecedentedly precise cosmic microwave background (CMB) data are expected
from ongoing and near-future CMB Stage-III and IV surveys, which will yield
reconstructed CMB lensing maps with effective resolution approaching several
arcminutes. The small-scale CMB lensing fluctuations receive non-negligible
contributions from nonlinear structure in the late-time density field. These
fluctuations are not fully characterized by traditional two-point statistics,
such as the power spectrum. Here, we use $N$-body ray-tracing simulations of
CMB lensing maps to examine two higher-order statistics: the lensing
convergence one-point probability distribution function (PDF) and peak counts.
We show that these statistics contain significant information not captured by
the two-point function, and provide specific forecasts for the ongoing
Stage-III Advanced Atacama Cosmology Telescope (AdvACT) experiment. Considering
only the temperature-based reconstruction estimator, we forecast 9$\sigma$
(PDF) and 6$\sigma$ (peaks) detections of these statistics with AdvACT. Our
simulation pipeline fully accounts for the non-Gaussianity of the lensing
reconstruction noise, which is significant and cannot be neglected. Combining
the power spectrum, PDF, and peak counts for AdvACT will tighten cosmological
constraints in the $\Omega_m$-$\sigma_8$ plane by $\approx 30\%$, compared to
using the power spectrum alone.
