  This article studies the Gram random matrix model $G=\frac1T\Sigma^{\rm
T}\Sigma$, $\Sigma=\sigma(WX)$, classically found in the analysis of random
feature maps and random neural networks, where $X=[x_1,\ldots,x_T]\in{\mathbb
R}^{p\times T}$ is a (data) matrix of bounded norm, $W\in{\mathbb R}^{n\times
p}$ is a matrix of independent zero-mean unit variance entries, and
$\sigma:{\mathbb R}\to{\mathbb R}$ is a Lipschitz continuous (activation)
function --- $\sigma(WX)$ being understood entry-wise. By means of a key
concentration of measure lemma arising from non-asymptotic random matrix
arguments, we prove that, as $n,p,T$ grow large at the same rate, the resolvent
$Q=(G+\gamma I_T)^{-1}$, for $\gamma>0$, has a similar behavior as that met in
sample covariance matrix models, involving notably the moment
$\Phi=\frac{T}n{\mathbb E}[G]$, which provides in passing a deterministic
equivalent for the empirical spectral measure of $G$. Application-wise, this
result enables the estimation of the asymptotic performance of single-layer
random neural networks. This in turn provides practical insights into the
underlying mechanisms into play in random neural networks, entailing several
unexpected consequences, as well as a fast practical means to tune the network
hyperparameters.
