Localization due to topological stochastic disorder in active networks

An active network is a prototype model in nonequilibrium statistical mechanics. It can represent, for example, a system with particles that have a self-propulsion mechanism. Each node of the network specifies a possible location of a particle and its orientation. The orientation (which is formally like a spin degree of freedom) determines the self-propulsion direction. The bonds represent the possibility to make transitions: to hop between locations or to switch the orientation. In systems of experimental interest (Janus particles), the self-propulsion is induced by illumination. An emergent aspect is the topological stochastic disorder (TSD). It is implied by the nonuniformity of the illumination. In technical terms the TSD reflects the local nonzero circulations (affinities) of the stochastic transitions. This type of disorder, unlike a nonhomogeneous magnetic field, is non-Hermitian and can lead to the emergence of a complex relaxation spectrum. It is therefore dramatically distinct from the conservative Anderson-type or Sinai-type disorder. We discuss the consequences of having TSD. In particular we illuminate three different routes to underdamped relaxation and show that localization plays a major role in the analysis. Implications of the bulk-edge correspondence principle are addressed too.