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Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2007

The fixed-trace β-Hermite ensemble of random matrices and the low temperature distribution of the determinant of an N × N β-Hermite matrix

Résumé

The β-Hermite ensemble (β-HE) of tridiagonal N × N random matrices of Dumitriu and Edelman (2002 J. Math. Phys. 43 5830) is a continuum of ensembles in which β, the reciprocal of the temperature in the 2D electrostatic interpretation of the eigenvalue characteristics, can take any value. The eigenvalue distributions coincide with those of the classical Gaussian ensembles (GOE, GUE, GSE) for β = 1, 2, 4. A fixed-trace β-Hermite ensemble (β-FTHE) is defined from the β-HE and is used to extend the spherical ensembles of classical symmetries to β-spherical ensembles. At low temperature, when β → ∞, for a fixed value of N, the asymptotic distributions of reduced determinants DN,β of random N × N β-H and β-FTH matrices are shown to be standard Gaussians. Accordingly, the fluctuations of the potential at the origin, -ln |DN,β|, have a generalized Gumbel distribution at low temperature. For large N and large β, a ln(N) variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions.

Dates et versions

hal-00906008 , version 1 (19-11-2013)

Identifiants

Citer

Gérard Le Caër, Renaud Delannay. The fixed-trace β-Hermite ensemble of random matrices and the low temperature distribution of the determinant of an N × N β-Hermite matrix. Journal of Physics A: Mathematical and Theoretical, 2007, 40, pp.1561-1584. ⟨10.1088/1751-8113/40/7/009⟩. ⟨hal-00906008⟩
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