Infinite-cluster geometry in central-force networks

TL;DR

This paper investigates the structure of infinite clusters in central-force networks, revealing a fractal backbone and finite-volume dangling ends, and characterizes their phase transition behavior near the rigidity threshold.

Contribution

It introduces a new mean field theory for the backbone exponent and provides simulation results that refine the understanding of the phase transition in such networks.

Findings

The infinite cluster comprises a fractal backbone and finite-volume dangling ends.

A first-order transition occurs in the infinite cluster density at the rigidity threshold.

Simulation results yield a backbone exponent Beta' of approximately 0.255.

Abstract

We show that the infinite percolating cluster (with density P_inf) of central-force networks is composed of: a fractal stress-bearing backbone (Pb) and; rigid but unstressed ``dangling ends'' which occupy a finite volume-fraction of the lattice (Pd). Near the rigidity threshold pc, there is then a first-order transition in P_inf = Pd + Pb, while Pb is second-order with exponent Beta'. A new mean field theory shows Beta'(mf)=1/2, while simulations of triangular lattices give Beta'_tr = 0.255 +/- 0.03.