A smooth transition from Wishart to GOE. (English) Zbl 1447.60027
Summary: It is well known that an \(n \times n\) Wishart matrix with \(d\) degrees of freedom is close to the appropriately centered and scaled Gaussian orthogonal ensemble (GOE) if \(d\) is large enough. Recent work of S. Bubeck et al. [Random Struct. Algorithms 49, No. 3, 503–532 (2016; Zbl 1349.05315)], and independently T. Jiang and D. Li [J. Theor. Probab. 28, No. 3, 804–847 (2015; Zbl 1333.15032)], shows that the transition happens when \(d = \Theta ( n^{3} )\). Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when \(d / n^{3} \rightarrow c \in (0, \infty )\). This shows, in particular, that the phase transition from Wishart to GOE is smooth.
MSC:
| 60B20 | Random matrices (probabilistic aspects) |
Keywords:
random matrix theory; Wishart distribution; Gaussian orthogonal ensemble; total variation; phase transitionReferences:
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