Discrete random structures: enumeration and scaling limits

This research network is on random discrete structures, which are ubiquitous in many areas of modern mathematics, and are essential to describe a host of phenomena in mathematical physics. For instance they play a key role in our understanding of phase transitions, which describe how physical systems can undergo abrupt changes (as when water changes from liquid to solid state below 0C). We will in particular focus on a number of fundamental two-dimensional examples, including the celebrated dimer model and planar graphs. By combining probabilistic and combinatorial perspectives we will aim to answer some of the most basic questions about these models: How to enumerate them, either exactly or approximately? How can one describe their random geometry in the large scale limit? How to explain that these structures keep arising under different guises in different problems? Such questions also have deep connections to questions in mathematical physics, from topological phase transitions to Liouville quantum gravity, which we aim to investigate.