TY - JOUR
T1 - Global Spectrum Fluctuations for Gaussian Beta Ensembles
T2 - A Martingale Approach
AU - Trinh, Khanh Duy
N1 - Funding Information:
The author would like to thank the referee for many useful comments. This work is supported by JSPS KAKENHI Grant No. JP16K17616.
Funding Information:
The author would like to thank the referee for many useful comments. This work is supported by JSPS KAKENHI Grant No. JP16K17616.
Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.
PY - 2019/9/1
Y1 - 2019/9/1
N2 - The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter β is allowed to vary with the matrix size n. In particular, we show that as n→ ∞ with nβ→ ∞, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.
AB - The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter β is allowed to vary with the matrix size n. In particular, we show that as n→ ∞ with nβ→ ∞, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.
KW - Gaussian beta ensembles
KW - Martingale difference central limit theorem
KW - Semicircle law
KW - Tridiagonal random matrices
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U2 - 10.1007/s10959-017-0794-9
DO - 10.1007/s10959-017-0794-9
M3 - Article
AN - SCOPUS:85032020525
SN - 0894-9840
VL - 32
SP - 1420
EP - 1437
JO - Journal of Theoretical Probability
JF - Journal of Theoretical Probability
IS - 3
ER -