Given a graph G, its k-core is the largest subgraph of G whose vertices have degree at least k. If G is a random graph, the size of its k-core jumps discontinuously from zero to a finite fraction of G, as the number of edges in G increases. This phenomenon shares properties both of first and second order phase transitions, and appears (in slightly different formulations) in a large variety of problems, ranging from combinatorics, to computer science and coding theory. In statistical physics, it plays a crucial role in several mean field models for the glass transition.

I will explain how to compute exact finite size scaling properties of this transition and discuss its universality properties.

LPT ENS