Norm Convergence Rate for Multivariate Quadratic Polynomials of Wigner Matrices

We study Hermitian non-commutative 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 $q$ converges to a deterministic limit with a rate of convergence of $N^{-2\mathrm{/}3+o(1)}$. 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.