Evidences of conformal invariance in 2d rigidity percolation

TL;DR

This paper demonstrates that 2D rigidity percolation clusters are conformally invariant at criticality and uses conformal field theory to analyze universal finite size effects, revealing unexpected similarities with connectivity percolation.

Contribution

It provides the first numerical evidence of conformal invariance in 2D rigidity percolation clusters and applies conformal field theory to predict finite size effects.

Findings

Rigidity clusters are conformally invariant at criticality.

Universal finite size effects are predicted by conformal field theory.

Similarities between rigidity and connectivity percolation universality classes are observed.

Abstract

The rigidity transition occurs when, as the density of microscopic components is increased, a disordered medium becomes able to transmit and ensure macroscopic mechanical stability, owing to the appearance of a space-spanning rigid connected component, or cluster. As a continuous phase transition it exhibits a scale invariant critical point, at which the rigid clusters are random fractals. We show, using numerical analysis, that these clusters are also conformally invariant, and we use conformal field theory to predict the form of universal finite size effects. Furthermore, although connectivity and rigidity percolation are usually though to belong to different universality classes and thus be of fundamentally different natures, we provide evidence of unexpected similarities between the statistical properties of their random clusters at criticality. Our work opens a new research avenue through the application of the powerful 2D conformal field theory tools to understand the critical behavior of a wide range of physical and biological materials exhibiting such a mechanical transition.