  We study an "inner-product kernel" random matrix model, whose empirical
spectral distribution was shown by Xiuyuan Cheng and Amit Singer to converge to
a deterministic measure in the large $n$ and $p$ limit. We provide an
interpretation of this limit measure as the additive free convolution of a
semicircle law and a Marcenko-Pastur law. By comparing the tracial moments of
this random matrix to those of a deformed GUE matrix with the same limiting
spectrum, we establish that for odd kernel functions, the spectral norm of this
matrix convergences almost surely to the edge of the limiting spectrum. Our
study is motivated by the analysis of a covariance thresholding procedure for
the statistical detection and estimation of sparse principal components, and
our results characterize the limit of the largest eigenvalue of the thresholded
sample covariance matrix in the null setting.
