Local laws describe the phenomenon that eigenvalues of random matrices converge to deterministic densities on scales just slightly above the typical eigenvalue spacing as the dimension of the matrices tends to infinity. This thesis explores both the nature of such limiting densities as well as the local laws that describe the convergence to these limiting densities for specific ensembles of both Hermitian and non-Hermitian random matrices.

The class of random matrices we are studying in the Hermitian realm are noncommutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root growth at its edges and prove an optimal local law around these edges. Combining these two results, we establish that, as the dimension N of the matrices grows to infinity, the operator norm of such polynomials converges to a deterministic limit with a rate of convergence of N −2/3+o(1). Here, the exponent in the rate of convergence is optimal. For the specific reducible cases, we also provide a classification of all possible edge behaviours.

For non-Hermitian matrices X and Y with centered, independent and identically distributed (i.i.d.) entries, it has been shown that the eigenvalues of the rational function Y−1X converge to a uniform density after being projected onto the Riemann sphere. We provide a simple proof of a local law for this ensemble by reducing the problem to the local circular law, a well-established result that the eigenvalues of a single centered i.i.d. matrix converge to the uniform distribution on the complex unit disc on all scales larger than the typical eigenvalue spacing.