Elastic Backbone Defines a New Transition in the Percolation Model

The elastic backbone is the set of all shortest paths. We found a new phase transition at p(eb) above the classical percolation threshold at which the elastic backbone becomes dense. At this transition in 2D, its fractal dimension is 1.750 +/- 0.003, and one obtains a novel set of critical exponents beta(eb) = 0.50 +/- 0.02, gamma(eb) = 1.97 +/- 0.05, and nu(eb) = 2.00 +/- 0.02, fulfilling consistent critical scaling laws. Interestingly, however, the hyperscaling relation is violated. Using Binder's cumulant, we determine, with high precision, the critical probabilities p(eb) for the triangular and tilted square lattice for site and bond percolation. This transition describes a sudden rigidification as a function of density when stretching a damaged tissue.