Universality for the global spectrum of random inner-product kernel matrices in the polynomial regime

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(XTX), where X ∈ ℝd × 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 dℓ for some ℓ > 0, not just the linear regime where ℓ = 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 ℓ is an integer (the linear regime ℓ = 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 Marvcenko-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 ℓ, the Marvcenko-Pastur term disappears (its weight in the free convolution vanishes), and the spectrum is just semicircular.