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Sylvester Index of Random Hermitian Matrices

Author

Listed:
  • Mohamed Bouali

    (Université de Tunis& Université de Tunis El Manar
    Sorbonne Université)

  • Jacques Faraut

    (Université de Tunis& Université de Tunis El Manar
    Sorbonne Université)

Abstract

The Sylvester index of a random Hermitian matrix in the Gaussian ensemble has been considered by Dean and Majumdar. We consider this Sylvester index for a matrix ensemble of random Hermitian matrices defined by a probability density of the form $$\exp \bigl (-\textrm{tr}\, Q(x))\bigr )$$ exp ( - tr Q ( x ) ) ) , where Q is a convex polynomial. The main result is the determination of the statistical distribution of the eigenvalues under the condition of a prescribed Sylvester index. We revisit some known results, giving complete proofs, for which we use logarithmic potential theory and complex analysis.

Suggested Citation

  • Mohamed Bouali & Jacques Faraut, 2024. "Sylvester Index of Random Hermitian Matrices," Journal of Theoretical Probability, Springer, vol. 37(1), pages 768-813, March.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:1:d:10.1007_s10959-022-01232-7
    DOI: 10.1007/s10959-022-01232-7
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