Elastic backbone defines a new transition in the percolation model

TL;DR

This paper identifies a new phase transition in percolation models where the elastic backbone becomes dense, revealing unique critical exponents and a violation of hyperscaling, with implications for understanding tissue rigidity.

Contribution

It introduces a novel phase transition at p_{eb} where the elastic backbone densifies, providing new critical exponents and insights into percolation theory.

Findings

Fractal dimension at transition: 1.750±0.003

Critical exponents: β_{eb}=0.50±0.02, γ_{eb}=1.97±0.05, ν_{eb}=2.00±0.02

Hyperscaling relation is violated

Abstract

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 $1.750\pm 0.003$, and one obtains a novel set of critical exponents $\beta_{eb} = 0.50\pm 0.02$, $\gamma_{eb} = 1.97\pm 0.05$, and $\nu_{eb} = 2.00\pm 0.02$ 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.