Eigenvalue distribution of the Hadamard product of sample covariance matrices in a quadratic regime

In this note, we prove that if $X\in {\mathbb{R}}^{n\times d}$ and $Y\in {\mathbb{R}}^{n\times p}$ are two independent matrices with i.i.d entries then the empirical spectral distribution of $\frac{1}{d}X{X}^{\mathrm{\top }}\odot \frac{1}{p}Y{Y}^{\mathrm{\top }}$, where $\odot$ denotes the Hadamard product, converges to the Marchenko–Pastur distribution of shape $\gamma$ in the quadratic regime of dimension $\frac{n}{dp}\to \gamma$ and $\frac{p}{d}\to a$.