Abstract
The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter \(\beta \) is allowed to vary with the matrix size n. In particular, we show that as \(n \rightarrow \infty \) with \(n\beta \rightarrow \infty \), the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.
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Acknowledgements
The author would like to thank the referee for many useful comments. This work is supported by JSPS KAKENHI Grant No. JP16K17616.
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Trinh, K.D. Global Spectrum Fluctuations for Gaussian Beta Ensembles: A Martingale Approach. J Theor Probab 32, 1420–1437 (2019). https://doi.org/10.1007/s10959-017-0794-9
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DOI: https://doi.org/10.1007/s10959-017-0794-9
Keywords
- Gaussian beta ensembles
- Tridiagonal random matrices
- Semicircle law
- Martingale difference central limit theorem