Phases in large combinatorial systems

Abstract

This is a status report on a companion subject to extremal combinatorics, obtained by replacing extremality properties with emergent structure, 'phases'. We discuss phases, and phase transitions, in large graphs and large permutations, motivating and using the asymptotic formalisms of graphons for graphs and permutons for permutations. Phase structure is shown to emerge using entropy and large deviation techniques.

A symmetry breaking transition in the edge/triangle network model

“…A phase is then a region of constraint values in which all these global observables vary smoothly with the constraint values, while transitions occur when the global state changes abruptly. (See [5,6,17,18,16,7] and the survey [15].) Our main focus is on the transition between two particular phases, in the system with the two constraints of edge and triangle density: phases in which the global states have different symmetry.…”

Untitled

“…Asymptotic structure results are often very useful for proving enumerative and probabilistic versions of the corresponding extremal problem. For example, the more general problem of understanding the structure of graphons with any given edge and r-clique densities appears in the study of exponential random graphs (see Chatterjee and Diaconis [4] and its follow-up papers), phases in large graphs (see the survey by Radin [25]), and large deviation inequalities for the clique density (see the survey by Chatterjee [3]). Last but not least, asymptotic structure results often greatly help, as a first step, in obtaining the exact structure of extremal graphs via the so-called stability approach pioneered by Simonovits [30].…”

The phases of large networks with edge and triangle constraints

“…The study of large dense graphs uses the mathematical tool of graphons, which we now review [6], making use of the discussion in [12]. We let G n denote the set of graphs on n nodes, which we label {1, .…”