Combinatorial Models of Rigidity and Renormalization
Résumé
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 of the parameters.
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