Noisy uncoupled chaotic map ensembles violate the law of large numbers

An ensemble of uncoupled chaotic maps with spatially synchronized parametric fluctuations violates the law of large numbers. This is clearly evident in the nonstatistical features of the mean field, whose mean-square deviation (MSD) does not fall as 1/N with increasing N, where N is the number of elements in the system. In fact the MSD saturates after a critical value of N=${\mathit{N}}_{\mathit{c}}$. This amazing phenomenon is reminiscent of the nonstatistical behavior in globally coupled chaos. Interestingly though, there is no coupling in this system, and so the emergence of a subtle coherence in the global dynamics, as suggested by the existence of size-independent fluctuations, is very intriguing.