Effect of the transition of networks from floppy to rigid on the diffusion coefficient

We investigate a tracer particle moving on a two-dimensional square lattice created by network formers (NF). The positions of all network formers are randomly displaced by a small amount from the nodes of the network. Each NF can be in a "floppy" or "rigid" state, depending on the number of bonds connecting it to neighboring network formers. The NF that have more than a specified number of m bonds are in rigid state, the remaining ones are in a floppy state. The energy of the tracer particle depends on its distance from those of the four nearest NF that are in "rigid" state. The NF in floppy state do not contribute to the energy. We here demonstrate that the a priori increase in the diffusion coefficient with the concentration of the floppy states goes through a crossover point, after which the increase is much sharper.