Rigidity percolation on the square lattice

Abstract

The square lattice with central forces between nearest neighbors is isostatic with a subextensive number of floppy modes. It can be made rigid by the random addition of next-nearest neighbor bonds. This constitutes a rigidity percolation transition which we study analytically by mapping it to a connectivity problem of two-colored random graphs. We derive an exact recurrence equation for the probability of having a rigid percolating cluster and solve it in the infinite volume limit. From this solution we obtain the rigidity threshold as a function of system size, and find that, in the thermodynamic limit, there is a mixed first-order-second-order rigidity percolation transition at the isostatic point.

6 pages, 3 figures

References in corpus (4)

• Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
• Elasticity and Response in Nearly Isostatic Periodic Lattices
• Algorithms for 3D rigidity analysis and a first order percolation transition
• Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation

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