Article,
Rigidity percolation in a field
Physical Review E, 68 (5): 056104+ (November 2003)
DOI: 10.1103/physreve.68.056104
Abstract
Rigidity percolation with g degrees of freedom per site is analyzed on randomly diluted Erdös-Renyi graphs, with average connectivity γ, in the presence of a field h. In the (γ,h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities nF of uncanceled degrees of freedom and γr of redundant bonds. Upon crossing the coexistence line, γr and nF are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely, that the density of uncanceled degrees of freedom is a ” free energy” for rigidity percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g+1) core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point γc, Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g+1) rigid core, i.e., the average coordination of these subgraphs is exactly 2g, although γc, the average coordination of the whole system, is smaller than 2g. γc is found to converge to 2g for large g, i.e., in this limit Maxwell counting is exact globally as well.
