Large scale structures typically consist of a large number of mutually interacting components and the collective behavior of this huge number of degrees of freedom leads to a host of intriguing phenomena: anomalous fluctuations, phase transitions, complex time evolutions and metastable behavior. By phase transition, a key concept for the proposal, we mean here an abrupt, qualitative change in the properties of the model when its parameters are changed. Mathematically, phase transitions are signaled by the appearance of different probability measures describing the possible states of the system but they are also signaled by a dramatic change in the convergence time of certain Markov chain Monte Carlo algorithms that are used to sample from the probability distribution of the sample. Systems of this type are well known and studied in statistical physics, but in the last decade their importance has been recognized in other frontier areas of scientific research, such as biology, genetics or theoretical computer science. At the same time it is widely recognized that a deep mathematical understanding of the behavior of these systems is essential. The latter requires the development and use of powerful probabilistic ideas and techniques. Probability is, in fact, the basis of a common language, that allows the sharing, adaptation and combination of key notions and techniques from the different areas involved. Around these topics and guided by the PI a small group of young and brilliant researchers formed in the last years and later developed into a somewhat larger team whose members are all very active and tightly collaborating, with already several important results which opened new directions and perspectives. I propose to build up on these advances by gathering around the host institution the team and guide it to further research on the interplay between equilibrium and dynamical aspects of phase transitions in large-scale structures.