ИСТИНА

Study of phase transition and the structure of equilibrium phases plays the central role in statistical mechanics of condensed systems. Indeed, often understanding the symmetries of the phases and proper choice of an order parameter is a crucial step allowing one to get an insight into what is actually going on in the system under consideration. Large equilibrium random networks are, clearly, an object very well suitable to be studied by the methods of statistical mechanics, and importing our insights into phase transition theory can be very instructive. In my talk I will give an overview of several examples where such import of the ideas from the theory of phase transitions turned out to be useful, and then concentrate on a particular example which we have been studying at length lately. Consider a large annealed network with a frozen degree distribution of nodes (that is to say, links between nodes can rewire, but the degree of each node always stays the same), and introduce a three-node interaction in this system which favours creation of closed triangles of bonds. We show that with increasing strength of the interaction, this system undergoes a first order phase transition from a homogeneous phase with locally almost tree-like structure and small concentration of triangles to a "triangle condensate" where the concentration of closed triangles drastically increases. Moreover, the condensed phase has a beautiful microstructure consisting: it is a set of almost fully-connected clicks with very small number of links between them. Such a microstructure formation can be easily understood and is akin to formation of microstructured phases in soft condensed matter.