Combinatorial models of rigidity and renormalization

Abstract

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 $k\le l<2k$. 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.

22 pages, 6 figures

References in corpus (5)

• Stressed backbone and elasticity of random central-force systems
• Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe Lattices
• Comparison of rigidity and connectivity percolation in two dimensions
• Hierarchical models of rigidity percolation
• Renormalization group treatment of rigidity percolation

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• The Chiral Potts Spin Glass in d=2 and 3 Dimensions
• Stepwise Positional-Orientational Order and the Multicritical-Multistructural Global Phase Diagram of the s=3/2 Ising Model from Renormalization-Group Theory