Expansion in supercritical random subgraphs of expanders and its consequences

In 2004, Frieze, Krivelevich and Martin established the emergence of a giant component in random subgraphs of pseudo-random graphs. We study several typical properties of the giant component, most notably its expansion characteristics. We establish an asymptotic vertex expansion of connected sets in the giant by a factor of (Formula presented.). From these expansion properties, we derive that the diameter of the giant is whp (Formula presented.), and that the mixing time of a lazy random walk on the giant is asymptotically (Formula presented.). We also show similar asymptotic expansion properties of (not necessarily connected) linear-sized subsets in the giant, and the typical existence of a large expander as a subgraph.