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

Abstract It has been recently noted that for a class of dynamical systems with explicit conservation laws represented via projector operators, the dynamics can be understood in terms of lower dimensional equations. This is the case, for instance, of memristive circuits. Memristive systems are important classes of devices with wide-ranging applications in electronic circuits, artificial neural networks, and memory storage. We show that such mean-field theories can emerge from averages over the group of orthogonal matrices, interpreted as cycle-preserving transformations applied to the projector operator describing Kirchhoff's laws. Our results provide insights into the fundamental principles underlying the behavior of resistive and memristive circuits and highlight the importance of conservation laws for their mean-field theories. In addition, we argue that our results shed light on the nature of the critical avalanches observed in quasi-two dimensional nanowires as boundary phenomena.