Renormalization group treatment of rigidity percolation

Abstract

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 $\beta =0.0775$ associated with the spanning rigid cluster and also for $d\nu =3.533$ 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 $\alpha$ and establish hyperscaling. The exact analytical procedures demonstrated on the chosen example readily generalize to wider classes of hierarchical lattice.

4 pages, 5 figures

Cited by in corpus (1)

• Combinatorial models of rigidity and renormalization