This is an announcement for the paper "Numerical range for random matrices" by Benoit Collins, Piotr Gawron, Alexander E. Litvak, and Karol Zyczkowski.

Abstract: We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. We show that the numerical range of complex Ginibre ensemble converges to the disk of radius $\sqrt{2}$. Since the spectrum of non-hermitian random matrices from the Ginibre ensemble lives asymptotically in a neighborhood of the unit disk, it follows that the outer belt of width $\sqrt{2}-1$ containing no eigenvalues can be seen as a quantification the non-normality of the complex Ginibre random matrix. We also show that the numerical range of upper triangular Gaussian matrices converges to the same disk of radius $\sqrt{2}$, while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to $\sqrt{2e}$.

Archive classification: math.OA math.FA math.PR quant-ph

Mathematics Subject Classification: 5A60, 47A12, 15B52 (primary), 46B06, 60B20 (secondary)

Remarks: 22 pages, 4 figures

Submitted from: gawron@iitis.pl

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1309.6203

or

http://arXiv.org/abs/1309.6203