Laboratoire de Physique Théorique
Toulouse - UMR 5152

In many systems of interacting agents, like spins, masses but also people, the interaction can be modelled though a potential whose decay gives precise information on the range of interaction. For instance, we experience every day the effect of a long-range force, gravity. Long-range interactions lead to, in the statistical mechanics frame, a very rich phenomenology of collective behaviours, like phase transitions and transient quasi stationary states. Now, if we take a dynamical system evolving on a network, this clear distinction between short-range and long-range interaction is completely blurred : we do not have anymore a range of interaction whose width is given by the potential decay ; in its place we have a mixture of short-range links and long-range ones. Moreover on a network, the metric distance can be less meaningful than the information distance. For instance, people are likely to have many friends in their physical neighborhood, but at the same time, they can also have acquaintances who are physically far away. This example hints to the core question of this work : in what topological conditions does long-range order arise ? It seems natural that the network structure can impact on the global behaviour of the system, but what is the dominant topological feature to steer it ? Of course the answer strongly depends on the dynamical system : in this seminar I will focus on some results for the classical XY−rotors model, showing how through the control on some network features we can indeed change the effective range of interaction and obtain a variety of dynamical behaviours.