Lower bounds and complex-eigenvalue distribution under strong convergence

Establish a lower bound on the spectral radius and prove convergence of the empirical complex eigenvalue distribution for non-self-adjoint noncommutative polynomials P(X^N,(X^N)*) under strong convergence hypotheses; equivalently, show that complex eigenvalue behavior is controlled in the same limit as operator norms for such polynomials.

Background

While strong convergence directly yields upper bounds on spectral radii for non-normal polynomials, lower bounds and distributional limits of complex eigenvalues remain poorly understood.

Progress in special cases (e.g., quadratic polynomials of Ginibre matrices) suggests the possibility of general results, but a broad theory aligning complex eigenvalue asymptotics with strong convergence is not yet available.