Limiting ESDs for sparse random circulant matrices over non-abelian groups

Determine whether the empirical spectral distributions of sparse random circulant matrices over the symmetric groups converge weakly in probability, and characterize their limiting behavior if convergence occurs.

Background

The main results concern sparse random circulant matrices indexed by finite abelian groups, whose simultaneous diagonalization by Fourier analysis makes their spectral behavior tractable. The paper proposes extending this investigation to non-abelian groups, where the commutative Fourier-analytic structure is no longer available in the same form.

The proposed intermediate problem specializes the non-abelian setting to symmetric groups and asks whether weak convergence in probability holds for the empirical spectral distributions. The subsequent discussion emphasizes that significant new ideas may be required in the non-abelian case.