  As we add rigid bars between points in the plane, at what point is there a
giant (linear-sized) rigid component, which can be rotated and translated, but
which has no internal flexibility? If the points are generic, this depends only
on the combinatorics of the graph formed by the bars. We show that if this
graph is an Erdos-Renyi random graph G(n,c/n), then there exists a sharp
threshold for a giant rigid component to emerge. For c < c_2, w.h.p. all rigid
components span one, two, or three vertices, and when c > c_2, w.h.p. there is
a giant rigid component. The constant c_2 \approx 3.588 is the threshold for
2-orientability, discovered independently by Fernholz and Ramachandran and
Cain, Sanders, and Wormald in SODA'07. We also give quantitative bounds on the
size of the giant rigid component when it emerges, proving that it spans a
(1-o(1))-fraction of the vertices in the (3+2)-core. Informally, the (3+2)-core
is maximal induced subgraph obtained by starting from the 3-core and then
inductively adding vertices with 2 neighbors in the graph obtained so far.
