Sylvester Index of Random Hermitian Matrices

Mohamed Bouali () and Jacques Faraut ()
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Mohamed Bouali: Université de Tunis& Université de Tunis El Manar
Jacques Faraut: Université de Tunis& Université de Tunis El Manar

Journal of Theoretical Probability, 2024, vol. 37, issue 1, 768-813

Abstract: 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.

Keywords: Eigenvalue; Random matrix; Sylvester index; Log gas method; 15A52; 41A29; 41A60 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-022-01232-7

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