Infinite-Cluster Geometry in Central-Force Networks

We show that the infinite percolating cluster (with density $P_{\infty }$) of central-force networks is composed of a fractal stress-bearing backbone ( $P_B$) and rigid but unstressed “isostatic ends” which occupy a finite volume fraction of the lattice ( $P_I$). Near the rigidity threshold $p_{*}$ there is then a first-order transition in $P_{\infty }{=P}_I{+P}_B$, while $P_B$ is second order with exponent ${\beta }^{\prime }$. A new mean-field theory suggests ${\beta }_{\mathrm{mf}}^{\prime }=1/2\mathrm{}$, while simulations of triangular lattices give ${\beta }_t^{\prime }=0.25\mathrm{}\pm 0.03$.