Conditions guaranteeing a linear-size supercritical component

Determine natural conditions on a finite graph G that guarantee that, whenever p cgeq (1+cepsilon)/clambda(G), the percolated graph G_p contains a component of order cOmega(|G|) with constant probability.

Background

The paper establishes that 1/clambda(G) is the critical probability for producing a component of order cOmega(clambda(G)) in an arbitrary finite graph, but this component need not contain a positive proportion of all vertices. The authors explain that the spectral radius may be localized in a small part of the graph: an induced subgraph on (1-o(1))|G| vertices can have substantially smaller spectral radius, causing that large portion of the graph to remain subcritical at p=(1+cepsilon)/clambda(G).

The unresolved problem is to identify natural structural or spectral hypotheses that overcome this localization issue and force a component of linear order in the supercritical percolated graph. The authors specifically caution that ordinary robust-connectivity assumptions may not suffice because they may fail to detect localization of the spectral radius.