In the phenomenon of emergence, a system of many interacting objects

exhibits the collective behavior of one or more “phases”, which are

only detectable or even meaningful for the system as a whole. This is

an organizing principle widely used in biology, physics and indeed all

the sciences: crystals, hurricanes, animal flocking etc. One wants to

understand the spontaneous appearance of phases in systems of large

size, in particular to determine a mechanism of some generality. A

convenient framework for such an analysis is large networks with

constraints. Such an analysis has been undertaken by the group of

Richard Kenyon, Charles Radin, Kui Ren and Lorenzo Sadun, on entropy singularities, the edge/triangle system I, the edge/triangle system II, multipodal structure,

order-disorder transitions and oversaturated networks. There is also a related asymptotic analysis of large permutations undertaken by the group of Richard Kenyon, Daniel Kral, Charles Radin and Peter Winkler: permutations with fixed pattern densities, and a review of phases in general combinatorial systems.