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

Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus