Renormalization group treatment of rigidity percolation

Renormalization group calculations are used to give exact solutions for rigidity percolation on hierarchical lattices. Algebraic scaling transformations for a simple example in two dimensions produce a transition of second order, with an unstable critical point and associated scaling laws. Values are provided for the order parameter exponent $β= 0.0775$ associated with the spanning rigid cluster and also for $d ν= 3.533$ which is associated with an anomalous lattice dimension $d$ and the divergence in the correlation length near the transition. In addition we argue that the number of floppy modes $F$ plays the role of a free energy and hence find the exponent $α$ and establish hyperscaling. The exact analytical procedures demonstrated on the chosen example readily generalize to wider classes of hierarchical lattice.

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