# The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

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## Abstract

Given an ensemble of *N*×*N* random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as *N*→∞. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric *m*-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period *m*. We view *m* as a “dial” we can “turn” from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities *f* _{ m } show a visually stunning convergence to the semi-circle as *m*→∞, which we prove.

In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that *f* _{ m } is the product of a Gaussian and a certain even polynomial of degree 2*m*−2; the formula is the same as that for the *m*×*m* Gaussian Unitary Ensemble (GUE). The proof is by derivation of the moments from the eigenvalue trace formula. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the *m*-block circulant pattern, dropping the assumption that the *m* random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an *m*-period, depending not only on the frequency at which each element appears, but also on the way the elements are arranged.

## Keywords

Limiting spectral measure Circulant and Toeplitz matrices Random matrix theory Convergence Method of moments Orientable surfaces Euler characteristic## Mathematics Subject Classification (2000)

15B52 60F05 11D45 60F15 60G57 62E20## Notes

### Acknowledgements

M. Koloğlu and G.S. Kopp were partially supported by Williams College and NSF Grants DMS0855257 and DMS0850577, and S.J. Miller was partly supported by NSF Grant DMS0970067. It is a pleasure to thank our colleagues from the Williams College 2010 SMALL REU program as well as the participants of the ICM Satellite Meeting in Probability & Stochastic Processes (Bangalore, 2010) for many helpful conversations, especially Arup Bose and Rajat Hazra. We would also like to thank Elizabeth Townsend Beazley for comments on Wentao Xiong’s senior thesis, which is the basis of Appendix B.

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