Spectral statistics of interpolating random circulant matrix and its applications to random circulant graphs

We consider a versatile matrix model of the form A+iB, where A and B are real random circulant matrices with independent but, in general, nonidentically distributed Gaussian entries. For this model, we derive exact results for the joint probability density function and find that it is a multivariate Gaussian. Arbitrary order marginal density therefore also readily follows. It is demonstrated that by adjusting the averages and variances of the Gaussian elements of A and B, we can interpolate between a remarkably wide range of eigenvalue distributions in the complex plane. In particular, we can examine the crossover between a random real circulant matrix and a random complex circulant matrix. We also extend our study to include Wigner-like and Wishart-like matrices constructed from our general random circulant matrix. To validate our analytical findings, Monte Carlo simulations are conducted, which confirm the accuracy of our results. Additionally, we compare our analytical results with the spectra of adjacency matrices from various random circulant graphs. Despite the difference in entry distributions-Gaussian in our model and non-Gaussian in the adjacency matrices-the densities show excellent agreement in the large-dimension limit.