Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus

Abstract We give a quantitative estimate for the quantum mean absolute deviation on hyperbolic surfaces of finite area in terms geometric parameters such as genus, number cusps and injectivity radius. It implies delocalisation result ergodicity type eigenfunctions Laplacian that Benjamini-Schramm converge to plane. show this is generic Mirzakhani’s model random chosen uniformly with respect Weil-Petersson volume. Depending particular sequence considered gives most cusp forms or Eisenstein series.