We study the probability distribution function of the long-time values of observables being timeevolved by Hamiltonians modeling clean and disordered one-dimensional chains of many spin-1/2 particles. In particular we analyze the return probability and its version for a completely extended initial state, the so-called spectral form factor. We complement our analysis with the spin autocorrelation and connected spin-spin correlation functions, both of interest in experiments with quantum simulators. We show that the distribution function has a universal shape provided the Central Limit Theorem holds. Explicitly, the shape is exponential for the return probability and spectral form factor, meanwhile it is Gaussian for the few-body observables. We also discuss implications over the so-called many-body localization. Remarkably, our approach requires only a single sample of the dynamics, which is quite advantageous for experiments and theory.[1] M. Clusel and E. Bertin, Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics, Int.
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