  We consider certain large random matrices, called random inner-product kernel
matrices, which are essentially given by a nonlinear function $f$ applied
entrywise to a sample-covariance matrix, $f(X^TX)$, where $X \in \mathbb{R}^{d
\times N}$ is random and normalized in such a way that $f$ typically has
order-one arguments. We work in the polynomial regime, where $N \asymp d^\ell$
for some $\ell > 0$, not just the linear regime where $\ell = 1$. Earlier work
by various authors showed that, when the columns of $X$ are either uniform on
the sphere or standard Gaussian vectors, and when $\ell$ is an integer (the
linear regime $\ell = 1$ is particularly well-studied), the bulk eigenvalues of
such matrices behave in a simple way: They are asymptotically given by the free
convolution of the semicircular and Mar\v{c}enko-Pastur distributions, with
relative weights given by expanding $f$ in the Hermite basis. In this paper, we
show that this phenomenon is universal, holding as soon as $X$ has i.i.d.
entries with all finite moments. In the case of non-integer $\ell$, the
Mar\v{c}enko-Pastur term disappears (its weight in the free convolution
vanishes), and the spectrum is just semicircular.
