Non-linear Random Matrix Theory (RMT) has recently emerged as a powerful paradigm for the theoretical understanding of deep learning theory. Throughout recent works, a universality principle, the \textit{Gaussian Equivalence Theorem} (GET), has become an indispensable tool allowing for the behavior of complex nonlinear neural networks to be understood through tractable linear kernel models. This thesis contributes to this emerging field, first by using the GET universality principle to derive a novel scaling law in Neural Tangent Kernel (NTK) regression, and second by studying the implications of this idealized linear equivalence on a high-dimensional nonlinearly separable dataset.