Large gaps between random eigenvalues

Valkó, Benedek; Virág, Bálint

Ann. Probab., Tome 38 (2010) no. 1, p. 1263-1279 / Harvested from Project Euclid

Résumé

We show that in the point process limit of the bulk eigenvalues of β-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size λ is given by ¶ (κ_β+o(1))λ^γ_β exp((−β/64)λ²+(β/8−1/4)λ) ¶ as λ→∞, where ¶ γ_β=1/4(β/2+2/β−3) ¶ and κ_β is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157–165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron–Martin–Girsanov transformation in stochastic calculus.

Publié le : 2010-05-15
Classification: Eigenvalues of random matrices, large deviation, β-ensembles, 60F10, 15B52

@article{1275486193,
     author = {Valk\'o, Benedek and Vir\'ag, B\'alint},
     title = {Large gaps between random eigenvalues},
     journal = {Ann. Probab.},
     volume = {38},
     number = {1},
     year = {2010},
     pages = { 1263-1279},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1275486193}
}

Valkó, Benedek; Virág, Bálint. Large gaps between random eigenvalues. Ann. Probab., Tome 38 (2010) no. 1, pp.  1263-1279. http://gdmltest.u-ga.fr/item/1275486193/