We consider the effects of long-range temporal correlations in many-particle systems, focusing particularly on fluctuations about the typical behaviour. For a specific class of memory dependence we discuss the modification of the large deviation principle describing the probability of rare currents and show how superdiffusive behaviour can emerge. We illustrate the general framework with detailed calculations for a memory-dependent version of the totally asymmetric simple exclusion process as well as indicating connections to other recent work.
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