Rigidity percolation in a field

Abstract

Rigidity Percolation with g degrees of freedom per site is analyzed on randomly diluted Erdos-Renyi graphs with average connectivity gamma, in the presence of a field h. In the (gamma,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 n_f of uncanceled degrees of freedom and gamma_r of redundant bonds. Upon crossing the coexistence line, n_f and gamma_r 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 gamma_c, 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 gamma_r, the average coordination of the whole system, is smaller than 2g. gamma_c is found to converge to 2g for large g, i.e. in this limit Maxwell counting is exact globally as well. This paper is dedicated to Dietrich Stauffer, on the occasion of his 60th birthday.

arXiv commentRevTeX4, psfig, 16 pages. Equation numbering corrected. Minor typos corrected

References in corpus (11)

• Stressed backbone and elasticity of random central-force systems
• The three-dimensional, three-state Potts Model in an External Field
• Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe Lattices
• Comparison of rigidity and connectivity percolation in two dimensions
• First-order rigidity transition on Bethe Lattices
• Rigorous analysis of discontinuous phase transitions via mean-field bounds
• Evidence of exactness of the mean field theory in the nonextensive regime of long-range spin models
• Cluster Percolation and Thermal Critical Behaviour
• Cluster Percolation and Critical Behaviour in Spin Models and SU(N) Gauge Theories
• Common features of deconfining and chiral critical points in QCD and the three state Potts model in an external field
• Cluster Percolation and Explicit Symmetry Breaking in Spin Models

Cited by in corpus (9)

• k-core organization of complex networks
• Inferring Network Mechanisms: The Drosophila melanogaster Protein Interaction Network
• Energy landscape and rigidity
• Relaxation and physical aging in network glasses: a review
• Heterogeneous-k-core versus Bootstrap Percolation on Complex Networks
• Rigidity percolation by next-nearest-neighbor braces on generic and regular isostatic lattices
• Finite temperature mechanical instability in disordered lattices
• Exactly solvable models of adaptive networks
• Two rigidity percolation transitions on binary Bethe networks and the intermediate phase in glass