In this paper, we show that the Lipschitz-Killing curvatures for the excursion sets of arithmetic random waves (toral Gaussian eigenfunctions) are dominated, in the high-frequency regime, by a single chaotic component. The latter can be written as a simple explicit function of the threshold parameter times the centered norm of these random fields; as a consequence, these geometric functionals are fully correlated in the high-energy limit. The derived formulae show a clear analogy with related results on the round unit sphere and suggest the existence of a general formula for geometric functionals of random eigenfunctions on Riemannian manifolds.
Cammarota, V., Marinucci, D., Rossi, M. (2023). Lipschitz-Killing Curvatures for Arithmetic Random Waves. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 24(2), 1095-1147 [10.2422/2036-2145.202010_065].
Lipschitz-Killing Curvatures for Arithmetic Random Waves
Rossi, Maurizia
2023
Abstract
In this paper, we show that the Lipschitz-Killing curvatures for the excursion sets of arithmetic random waves (toral Gaussian eigenfunctions) are dominated, in the high-frequency regime, by a single chaotic component. The latter can be written as a simple explicit function of the threshold parameter times the centered norm of these random fields; as a consequence, these geometric functionals are fully correlated in the high-energy limit. The derived formulae show a clear analogy with related results on the round unit sphere and suggest the existence of a general formula for geometric functionals of random eigenfunctions on Riemannian manifolds.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.