alysis on finite abelian groups which are used here and points out their immediate consequences for $G$-circulant matrices; some notation and conventions used in the remainder of the paper are established there. Section \[S:Gaussian\] investigates the spectra of some random $G$-circulant matrices whose entries are Gaussian random variables. The invariance properties of Gaussian random variables allow an easy detailed study to be undertaken which illuminates the general situation, in particular the role of the number of elements of order $2$. Finally, Section \[S:general\] determines the asymptotic behavior of the spectrum for general entries with finite variances.

The cases of $G$-circulant matrices with heavy-tailed entries, and of random $G$-circulant matrices when $G$ is a nonabelian finite group, will be investigated in future work.

Acknowledgements {#acknowledgements .unnumbered}
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The author thanks Persi Diaconis for encouragement and pointers to the literature, John Duncan for helpful discussions about character theory, and the referee for careful reading and useful comments. This research was partly supported by National Science Foundation grant DMS-0902203.

Some Fourier analysis and notation {#S:Fourier}
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For a finite abelian group $G$, we denote by $\widehat{G}$ the family of group homomorphisms $\chi : G \to {\mathbb{T}}$, where ${\mathbb{T}}$ is the multiplicative group $\{z \in {\mathbb{C}}\mid {\left\vert z \right\vert} = 1\}$. The elements of $\widehat{G}$ are called characters of $G$; $\widehat{G}$ is a group under the operation of pointwise multiplication. The multiplicative inverse of a character $\chi$ is its pointwise complex conjugate $\overline{\chi}$. From the homomorphism property it follows that for $a \in G$ and $\chi \in \widehat{G}$, $\chi(a^{-1}) =
\overline{\chi}(a)$.

We denote by $\ell^2(G)$ the space of functions $f: G \to {\mathbb{C}}$ equipped with the inner product $${\left\langle f, g \right\rangle} = \sum_{a \in G} f(a) \o