Ramanujan-strengthened strong convergence

Investigate whether there exists a strengthened form of strong convergence that, analogously to the Ramanujan property for regular graphs, yields exact (o(1)-free) operator norm bounds for noncommutative polynomials in suitable deterministic or random matrix families; specifically, determine whether one can formulate and prove a Ramanujan-type strong convergence principle.

Background

Standard strong convergence yields asymptotically sharp operator norm limits up to o(1) errors, while Ramanujan graphs achieve exact spectral bounds with no o(1). The authors raise whether an analogue of this exactness exists for strong convergence beyond the graph spectrum context.

Such a theory could unify optimal spectral gap phenomena with operator-norm convergence across polynomial functionals, but the existence of such a formulation is presently unclear.