State transition graph of the Preisach model and the role of return-point memory

The Preisach model has been useful as a null-model for understanding memory formation in periodically driven disordered systems. In amorphous solids for example, the athermal response to shear is due to localized plastic events (soft spots). As shown recently by one of us, the plastic response to applied shear can be rigorously described in terms of a directed network whose transitions correspond to one or more soft spots changing states. The topology of this graph depends on the interactions between soft-spots and when such interactions are negligible, the resulting description becomes that of the Preisach model. A first step in linking transition graph topology with the underlying soft-spot interactions is therefore to determine the structure of such graphs in the absence of interactions. Here we perform a detailed analysis of the transition graph of the Preisach model. We highlight the important role played by return point memory in organizing the graph into a hierarchy of loops and sub-loops. Our analysis reveals that the topology of a large portion of this graph is actually not governed by the values of the switching fields that describe the individual hysteretic behavior of the individual elements, but by a coarser parameter, a permutation $\rho$ which prescribes the sequence in which the individual hysteretic elements change their states as the main hysteresis loop is traversed. This in turn allows us to derive combinatorial properties, such as the number of major loops in the transition graph as well as the number of states $| \mathcal{R} |$ constituting the main hysteresis loop and its nested subloops. We find that $| \mathcal{R} |$ is equal to the number of increasing subsequences contained in the permutation $\rho$.