Limiting spectral laws for sparse random circulant matrices

Fix a positive integer $d$ and let $(G_n)_{n\geq1}$ be a sequence of finite abelian groups with orders tending to infinity. For each $n \geq 1$, let $C_n$ be a uniformly random $G_n$-circulant matrix with entries in $\{0,1\}$ and exactly $d$ ones in each row/column. We show that the empirical spectral distribution of $C_n$ converges weakly in expectation to a probability measure $\mu$ on $\mathbb{C}$ if and only if the distribution of the order of a uniform random element of $G_n$ converges weakly to a probability measure $\rho$ on $\mathbb{N}^*$, 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 $\rho$ is a Dirac mass $\delta_m$. In this case, $\mu$ is the $d$-fold convolution of the uniform distribution on the $m$-th roots of unity if $m\in\mathbb{N}$ or the unit circle if $m = \infty$. We also establish that, under further natural assumptions, the determinant of $C_n$ is $\pm\exp((c_{m,d}+o(1))|G_n|)$ with high probability, where $c_{m,d}$ is a constant depending only on $m$ and $d$.