For d-regular random G_n-circulant matrices, the ESD converges in expectation iff the order distribution of a uniform group element converges, and in probability iff that order law is a point mass.
Random non-Abelian G-circulant matrices
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Limiting spectral laws for sparse random circulant matrices
For d-regular random G_n-circulant matrices, the ESD converges in expectation iff the order distribution of a uniform group element converges, and in probability iff that order law is a point mass.