Absolute continuity of the limiting eigenvalue distribution of the random Toeplitz matrix

Arnab Sen (University of Cambridge)
Balint Virag (University of Toronto)

Abstract


We show that the limiting eigenvalue distribution of random symmetric Toeplitz matrices is absolutely continuous with density bounded by 8, partially answering a question of Bryc, Dembo and Jiang (2006). The main tool used in the proof is a spectral averaging technique from the theory of random Schrödinger operators. The similar question for Hankel matrices remains open

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 706-711

Publication Date: November 20, 2011

DOI: 10.1214/ECP.v16-1675

References

  1. Bhatia, R. Matrix analysis , Graduate Texts in Mathematics, Vol 169 (1997), Springer- Verlag, New York. Math. Review 1477662.
  2. Bryc, W., Dembo, A. and Jiang, T. Spectral measure of large random Hankel, Markov and Toeplitz matrices, Ann. Probab. 34(1) (2006), 1 - 38. Math. Review 2206341.
  3. Combes, J. M., Hislop, P. D. and Mourre, E. Spectral averaging, perturbation of singular spectra, and localization, Trans. Amer. Math. Soc. 348(12) (1996), 4883 - 4894. Math. Review 1344205.
  4. Sen, A. and Viràg B. The top eigenvalue of the random toeplitz matrix and the sine kernel (2011). available at Arxiv math 1109.5494.
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.