The full counting statistics encodes the probability distribution of a dynamical observable and is a dynamical analogue of the thermodynamic partition function. It is naturally discussed within the frameworks of the large deviation theory and the Lee-Yang theory of phase transitions, which we briefly review. By combining the two approaches we point out that, in the presence of dynamical critical point, a rich phenomenology of fluctuations is permissible. As an explicit demonstration we introduce an interacting cellular automaton, where an analytical computation of the full counting statistics is feasible. Asymptotic analysis of the exact solution gives access to the current distribution on all scales and explicit cumulant asymptotics, revealing, among other anomalous features, non-Gaussian typical fluctuations in equilibrium. The scaled cumulant generating function does not generate scaled cumulants. If time permits we will also discuss some recent results on anomalous fluctuations in the (anisotropic) Landau-Lifshitz model, a paradigmatic integrable model of interacting classical spins.

References:
Ž. Krajnik, J. Schmidt, V. Pasquier, E. Ilievski, T. Prosen. Exact anomalous current fluctuations in a deterministic interacting model. arXiv: 2201.05126
Ž. Krajnik, E. Ilievski, T. Prosen. Absence of Normal Fluctuations in an Integrable Magnet, arXiv:2109.13088