Phases in Large Combinatorial Systems

TL;DR

This paper explores the emergence of phase structures in large combinatorial systems like graphs and permutations, using entropy and large deviation methods to analyze phase transitions.

Contribution

It introduces a framework for understanding phases in large combinatorial systems through asymptotic formalisms like graphons and permutons, replacing extremality with emergent structure.

Findings

Phase structures emerge in large graphs and permutations.

Entropy and large deviation techniques reveal phase transitions.

The approach generalizes extremal combinatorics to emergent phenomena.

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.