Rigidity percolation on the square lattice

TL;DR

This paper investigates the rigidity percolation transition in a square lattice with added next-nearest neighbor bonds, revealing a mixed first-order-second-order transition at the isostatic point through analytical methods.

Contribution

It introduces an exact analytical approach to study the rigidity transition in the square lattice, mapping it to a two-colored random graph connectivity problem.

Findings

Identifies the rigidity threshold as a function of system size.

Shows a mixed first-order-second-order transition at the isostatic point.

Provides an exact recurrence equation for the probability of a rigid cluster.

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.