Consider the following simple process. Start with n points in the Euclidean plane, allowed to move freely. Now, uniformly at random, pick a pair of the points, and fix the distance between them, and then repeat this step. Eventually, the fixed distances cause subsets of the points to have Euclidean isometries as their only allowed motions, and the emergence of these “rigid components” has been studied, via simulation, in the physics community.
Shiva Kasiviswanathan, Cris Moore, and Louis Theran obtained a fairly detailed picture of the transition from flexibility to rigidity in random frameworks in their paper The Rigidity Transition in Random Graphs.
Louis has recently given talks on the subject at: