Haagerup-Thorbjørnsen's work on strong spectral convergence of Gaussian matrices II

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Abstract

We demonstrate that the Stieltjes transform of Hermitian random matrices NxN is the fixed point of a certain transformation (to within N^{-2}).

The study is completed using a method inspired by Froese-Hasler-Spitzer : we show that the transformation whose fixed point we are studying acts on Siegel half-space, and we use the geometry of this space to show that it is contracting. This shows that the Stieltjes transform at the level of NxN matrices is close to the true fixed point, which is the Stieltjes transform arising from semicircular laws. Strong convergence follows.

References

Case of GUE matrices :

• U. Haagerup, S. Thorbjørnsen. A new application of random matrices: Ext(C∗red(F2)) is not a group. Annals of Mathematics, 162 (2005), 711-77

The case of GOE and symplectic matrices :

• H. Schultz, Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases. Probab. Theory Relat. Fields 131, 261-309 (2005)

Further references :

• R. Froese, D. Hasler, W. Spitzer, Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs, Journal of Functional Analysis, Volume 230, Issue 1, 1 January 2006, Pages 184-221
• J. Mingo, R. Speicher, Free Probability and Random Matrices (Chapter 3). Fields Institute Monographs, Springer, 2017