ASYMPTOTICS OF THE INERTIA MOMENTS AND THE VARIANCE CONJECTURE IN SCHATTEN BALLS
Résumé
We study the limit, as the dimension goes to infinity, of the moments of the Hilbert-Schmidt norm of a uniformly distributed matrix in the p-Schatten ball, with entries in the real, complex or quaternionic field. We also consider the restriction to the space of self-adjoint matrices. We build on the connection with spectral asymptotics of β-ensembles to adapt some fluctuation results due to Bekerman, Leblé and Serfaty [8]. When p > 3, this allows us to obtain the next asymptotic order for ratios of q-inertia moments of p-Schatten balls of self-adjoint matrices, and to establish a strong version of the variance conjecture for these families of convex bodies.
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