The rigidity transition in random graphs

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 Erdős-Rényi 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.