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

TL;DR

The paper introduces a versatile random matrix model ${\bf H}={\bf A}+i{\bf B}$ with real circulant ${\bf A},{\bf B}$ and Gaussian entries, proving that the joint eigenvalue distribution is exactly multivariate Gaussian. This enables exact marginals and a tunable interpolation between real and complex circulant ensembles, with extensions to Wigner-like and Wishart-like constructions. The authors validate the theory through Monte Carlo simulations and apply it to adjacency spectra of random circulant graphs, showing strong agreement with the Gaussian model in the large-dimension limit despite non-Gaussian graph edges. The work offers a tractable framework for non-Hermitian circulant ensembles and provides insights into the spectral statistics of structured random graphs, with potential extensions to elliptic Ginibre and non-Gaussian variants.

Abstract

We consider a versatile matrix model of the form ${\bf A}+i {\bf B}$, where ${\bf A}$ and ${\bf B}$ 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 ${\bf A}$ and ${\bf B}$, 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.

Figures (15)

• Figure 1: Probability densities of individual ordered eigenvalues of ${\bf H}$ in the complex plane for $N=5$. The averages and standard deviations of independent Gaussian elements of matrices ${\bf A}$ and ${\bf B}$ are $(u_1,u_2,u_3,u_4,u_5;\sigma_1,\sigma_2,\sigma_3,\sigma_4,\sigma_5)=(2,9,-7,-19/2,-5/3;1,2,1/2,2/7,4/5)$ and $(v_1,v_2,v_3,v_4,v_5;\tau_1,\tau_2,\tau_3,\tau_4,\tau_5)=(4,8,-15/2,3,20/3; 6/5, 2/3, 3/4,4/7,3/5)$, respectively. The simulation results, obtained from an ensemble comprising 100 000 matrices, are shown as histograms, while the two-dimensional surfaces are based on analytical result given in Eq. \ref{['marginal-joint']}. The ordering of the eigenvalues has been indicated using the numbers above the histograms.
• Figure 2: Probability densities of (a) real and (b) imaginary parts of individual ordered eigenvalues of ${\bf H}$. The parameter values are the same as those in Fig. \ref{['fig1']}. The histograms depict the results obtained from numerical simulations, while the solid lines represent the analytical results. The numbers above the histograms indicate the ordering of the eigenvalues.
• Figure 3: Probability density of an unordered eigenvalue of ${\bf H}$ in the complex plane for $N=5$. Parameter values and presentation scheme are as in Fig. \ref{['fig1']}. In this case, the analytical result employed is Eq. \ref{['mainjpd_uo']}.
• Figure 4: Probability densities of (a) real and (b) imaginary parts of an unordered eigenvalue of ${\bf H}$, corresponding to the one shown in Fig. \ref{['fig1']}. The histograms are obtained from numerical simulations, while the solid lines are derived from Eqs. \ref{['md1uo']} and \ref{['md2uo']}.
• Figure 5: Probability density of an unordered eigenvalue of the matrix ${\bf W=HH^\dag}$ for $N=3$. The variances of independent zero-mean Gaussian elements of matrices ${\bf A}$ and ${\bf B}$ are $(\sigma_1,\sigma_2,\sigma_3)=(1,7/2,3/4)$ and $(\tau_1,\tau_2,\tau_3)=(4/3,2/3,9/2)$, respectively. The histogram has been obtained using numerical simulation comprising 20 000 matrices and the solid line is based on Eq. \ref{['circWisuo']}.