Comparison of Connectivity and Rigidity Percolation

Summary

The development of tree models and the use of new algorithms in numerical studies have revolutionized our understanding of the geometry of rigidity percolation. In particular we now know that on triangular lattices, the rigidity transition is second order, but in a different universality class to the connectivity case. In contrast on trees the rigidity transition is first order while the connectivity transition is second order. The geometry of rigidity percolation is different from that of connectivity percolation due to the requirement of multiple connectivity in the rigidity case. However this is clearly not enough to ensure that the transition becomes first order. Perhaps a deeper question is whether the infinite cluster breaks up into two or an infinite number of subclusters when a critical bond is removed. In the connectivity case, the answer is clearly two. In the rigidity case it is infinity on trees and difficult to analyze precisely on triangular lattices. This and a host of other questions remain unanswered in this interesting class of problems.