Percolation of frozen order in glassy combinatorial problems

Abstract

A local order parameter which is important in the analysis of phase transitions in frustrated combinatorial problems is the probability that a node is frozen in a particular state. There is a percolative transition when an infinite connected cluster of these frozen nodes emerges. In this contribution, we develop theories based on this percolation process and discuss its relation to conventional connectivity percolation and its generalisation to k-connectivity percolation. The emergence of frozen order may also be considered to be a form of constraint percolation (CP) which enables us to draw analogies with rigidity percolation and its associated matching problems. We show that very simple CP processes on Bethe lattices lead to the replica symmetric equations for KSAT, coloring and the Viana-Bray model.