Global Spectrum Fluctuations for Gaussian Beta Ensembles: A Martingale Approach

Khanh Duy Trinh ()
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Khanh Duy Trinh: Kyushu University

Journal of Theoretical Probability, 2019, vol. 32, issue 3, 1420-1437

Abstract: Abstract The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $$\beta $$ β is allowed to vary with the matrix size n. In particular, we show that as $$n \rightarrow \infty $$ n → ∞ with $$n\beta \rightarrow \infty $$ n β → ∞ , the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.

Keywords: Gaussian beta ensembles; Tridiagonal random matrices; Semicircle law; Martingale difference central limit theorem; Primary 60B20; Secondary 60F05 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-017-0794-9

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