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It has been recently noted that for a class of dynamical systems withexplicit conservation laws represented via projector operators the dynamics canbe understood in terms of lower dimensional equations This is the case forinstance of memristive circuits Memristive systems are important classes ofdevices with wide ranging applications in electronic circuits artificial neuralnetworks and memory storage We show that such mean field theories can emergefrom averages over the group of orthogonal matrices interpreted as cyclepreserving transformations applied to the projector operator describingKirchhoffs laws Our results provide insights into the fundamental principlesunderlying the behavior of resistive and memristive circuits and highlight theimportance of conservation laws for their meanf ield theories In addition weargue that our results shed light on the nature of the critical avalanchesobserved in quasi-two dimensional nanowires as boundary phenomena

Cycle equivalence classes, orthogonal Weingarten calculus, and the mean field theory of memristive systems