2025/04/18 by Beker, Adrian
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2504.13833
Fix a positive integer d and let (Gn)n≥1 be a sequence of finite abelian groups with orders tending to infinity. For each n ≥ 1, let Cn be a uniformly random Gn-circulant matrix with entries in \0,1\ and exactly d ones in each row/column. We show that the empirical spectral distribution of Cn converges weakly in expectation to a probability measure μ on ℂ if and only if the distribution of the order of a uniform random element of Gn converges weakly to a probability measure ρ on ℕ^*, the one-point compactification of the natural numbers. Furthermore, we show that convergence in expectation can be strengthened to convergence in probability if and only if ρ is a Dirac mass δm. In this case, μ is the d-fold convolution of the uniform distribution on the m-th roots of unity if m∈ℕ or the unit circle if m = ∞. We also establish that, under further natural assumptions, the determinant of Cn is ±exp((cm,d+o(1))|Gn|) with high probability, where cm,d is a constant depending only on m and d.