Limiting spectral laws for sparse random circulant matrices
Adrian Beker
TL;DR
This work analyzes the limiting spectral behavior of sparse random $G_n$-circulant matrices with exactly $d$ ones per row/column, linking weak convergence of the empirical spectral distribution to the asymptotic order distribution of a uniform random element of $G_n$. The authors develop a framework combining Fourier analysis on finite abelian groups and the Hermitisation method to obtain precise necessary and sufficient conditions for convergence in expectation and in probability: convergence in expectation occurs iff the order distribution $\rho_{G_n}$ converges to a measure $\rho$, yielding $\mu=\sum_m \rho(\{m\})\eta_m^{*d}$, while convergence in probability occurs iff $\rho_{G_n}$ converges to a Dirac mass $\delta_m$, yielding $\mu=\eta_m^{*d}$. They further establish determinant asymptotics, showing $\frac{1}{|G_n|}\log|\det(C_n)|$ concentrates at $c_{m,d}=\int \log|z|\,d\eta_m^{*d}(z)$ under a nondegeneracy condition. The analysis combines uniform log-integrability with a detailed study of sums of roots of unity and uses a Möbius-inversion argument to control convergence of the spectral law, with explicit results for when $G_n=\mathbb{Z}/n\mathbb{Z}$ or $G_n=(\mathbb{Z}/m\mathbb{Z})^n$. Overall, the paper provides a sharp description of how the underlying group structure governs the limiting spectral behavior of sparse circulant matrices and connects it to determinant growth, contributing to the understanding of structured random matrices and random Cayley graphs.
Abstract
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 $μ$ 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 $ρ$ 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 $ρ$ is a Dirac mass $δ_m$. In this case, $μ$ 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 $d$.