  We analytically compute the full counting statistics of charge transfer in a
classical automaton of interacting charged particles. Deriving a closed-form
expression for the moment generating function with respect to a stationary
equilibrium state, we employ asymptotic analysis to infer the structure of
charge current fluctuations for a continuous range of timescales. The solution
exhibits several unorthodox features. Most prominently, on the timescale of
typical fluctuations the probability distribution of the integrated charge
current in a stationary ensemble without bias is distinctly non-Gaussian
despite diffusive behavior of dynamical charge susceptibility. While inducing a
charge imbalance is enough to recover Gaussian fluctuations, we find that
higher cumulants grow indefinitely in time with different exponents, implying
singular scaled cumulants. We associate this phenomenon with the lack of a
regularity condition on moment generating functions and the onset of a
dynamical critical point. In effect, the scaled cumulant generating function
does not, irrespectively of charge bias, represent a faithful generating
function of the scaled cumulants, yet the associated Legendre dual yields the
correct large-deviation rate function. Our findings hint at novel types of
dynamical universality classes in deterministic many-body systems.
