Percolation on random graphs with a fixed degree sequence

We consider bond percolation on random graphs with given degrees and bounded average degree. In particular, we consider the order of the largest component after the random deletion of the edges of such a random graph. We give a rough characterisation of those degree distributions for which bond percolation with high probability leaves a component of linear order, known usually as a giant component. We show that essentially the critical condition has to do with the tail of the degree distribution. Our proof makes use of recent technique, which is based on the switching method and avoids the use of the classic configuration model on degree sequences that have a limiting distribution. Thus our results hold for sparse degree sequences without the usual restrictions that accompany the configuration model.