Extremal driving as a mechanism for generating long-term memory

Abstract

It is argued that systems whose elements are renewed according to an extremal criterion can generally be expected to exhibit long-term memory. This is verified for the minimal extremally driven model, which is first defined and then solved for all system sizes N\geq2 and times t\geq0, yielding exact expressions for the persistence R(t)=[1+t/(N-1)]^{-1} and the two-time correlation function C(t_{\rm w}+t,t_{\rm w})=(1-1/N)(N+t_{\rm w})/(N+t_{\rm w}+t-1). The existence of long-term memory is inferred from the scaling of C(t_{\rm w}+t,t_{\rm w})\sim f(t/t_{\rm w}), denoting {\em aging}. Finally, we suggest ways of investigating the robustness of this mechanism when competing processes are present.