The Freezing Threshold for k-Colourings of a Random Graph

We determine the exact value of the freezing threshold, rfk, for k-colourings of a random graph when k≥ 14. We prove that for random graphs with density above rfk, almost every colouring is such that a linear number of vertices are frozen, meaning that their colour cannot be changed by a sequence of alterations whereby we change the colours of o(n) vertices at a time, always obtaining another proper colouring. When the density is below rfk, then almost every colouring is such that every vertex can be changed by a sequence of alterations where we change O(log n) vertices at a time. Frozen vertices are a key part of the clustering phenomena discovered using methods from statistical physics. The value of the freezing threshold was previously determined by the nonrigorous cavity method.