Renormalization group treatment of rigidity percolation

TL;DR

This paper applies renormalization group methods to solve rigidity percolation on hierarchical lattices, revealing critical behavior and scaling laws for phase transitions in elastic networks.

Contribution

It provides exact solutions for rigidity percolation using renormalization group techniques on hierarchical lattices, including critical exponents and scaling relations.

Findings

Identified a second-order transition with specific critical exponents.

Derived scaling laws and hyperscaling relations for rigidity percolation.

Demonstrated the generalizability of the analytical approach.

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.