Combinatorial Models of Rigidity and Renormalization

We first introduce the percolation problems associated with the graph theoretical concepts of (k, l)-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization transformation for (k, l)-percolation problems, and investigate its domain of validity. In particular, we show that it allows an exact solution of (k, l)-percolation problems on hierarchical graphs, for k ≤ l < 2k. We introduce and solve by renormalization such a model, which has the interesting feature of showing both ordinary percolation and rigidity percolation phase transitions, depending on the values