What question did this study set out to answer?

To establish thin-shell concentration and Gaussian fluctuation properties for rotationally invariant random simplices, and to sharpen asymptotic distribution bounds for Gaussian random matrix determinants.

January 1, 2022Electronic Journal of ProbabilityOpen Access

Thin-shell theory for rotationally invariant random simplices

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Authors

JHJohannes HeinyUniversity of Copenhagen SJSamuel G. G. JohnstonUniversity College Dublin JPJoscha ProchnoUniversity of Passau

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Implication

Theoretical study reveals thin-shell concentration and Gaussian fluctuations in rotationally invariant random simplices, indicating sharpened determinant bounds for Gaussian random matrices.

Key Points

To establish thin-shell concentration and Gaussian fluctuation properties for rotationally invariant random simplices, and to sharpen asymptotic distribution bounds for Gaussian random matrix determinants.
Formulated high-dimensional rotationally invariant probability densities using variational principles to characterize Euclidean norm concentration.
Analyzed the logarithmic volume of simplices formed by independent random vectors across Gaussian, beta, and beta prime distribution classes.
Applied geometric-to-matrix determinant correspondences to evaluate the asymptotic distributions of n-by-n standard Gaussian random matrix determinants.
Demonstrated that the Euclidean norm of high-dimensional random vectors concentrates around a variational minimizer with order 1/sqrt(n) Gaussian fluctuations.
Proved that the logarithmic volume of rotationally invariant random simplices exhibits universal asymptotic Gaussian behavior.
Sharpened the normal approximation error bound for the log-determinants of Gaussian random matrices to order (log n)^(3/2) / sqrt(n), improving upon previous asymptotic rates.

Cite This Study

Heiny et al. (2022) studied this question.

synapsesocial.com/papers/6a7caad83dbea9de816115b7 https://doi.org/10.1214/21-ejp734

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Implication

Theoretical study reveals thin-shell concentration and Gaussian fluctuations in rotationally invariant random simplices, indicating sharpened determinant bounds for Gaussian random matrices.

Thin-shell theory for rotationally invariant random simplices

Discussion

Member takes

Implication

Key Points

Cite This Study

Thin-shell theory for rotationally invariant random simplices

Discussion

Member takes

Implication

Key Points

Cite This Study

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