Schramm-Loewner Evolution in 2D Rigidity Percolation

Amorphous solids may resist external deformation such as shear or compression, while they do not present any long-range translational order or symmetry at the microscopic scale. Yet, it was recently discovered that, when they become rigid, such materials acquire a high degree of symmetry hidden in the disorder fluctuations: their microstructure becomes statistically conformally invariant. In this Letter, we exploit this finding to characterize the universality class of central-force rigidity percolation (RP), using Schramm-Loewner evolution (SLE) theory. We provide numerical evidence that the interfaces of the mechanically stable structures (rigid clusters), at the rigidification transition, are consistently described by ${\mathrm{SLE}}_{\kappa }$, showing that this powerful framework can be applied to a mechanical percolation transition. Using well-known relations between different SLE observables and the universal diffusion constant $\kappa$, we obtain the estimation $\kappa \sim 2.9$ for central-force RP. This value is consistent, through relations coming from conformal field theory, with previously measured values for the clusters’ fractal dimension $D_f$ and correlation length exponent $\nu$, providing new, nontrivial relations between critical exponents for RP. These findings open the way to a fine understanding of the microstructure in other important classes of rigidity and jamming transitions.

Figure 1

Variance of the driving function ${\xi }_t$ as a function of time. Bottom right inset: rescaled probability density function of ${\xi }_t$ at four different times; in gray is the Gaussian distribution. Top left inset: quantile-quantile plots of ${\xi }_t$ versus $\mathcal{N}(0,1)$ at four different times. Gray lines have slopes $\sqrt{{\kappa }_{\text{driving}}t}$.

Figure 2

Variance of the winding angle $\theta$ as function of the distance $x$ along the curve. Top left inset: mean length $\mathbb{E}[l]$ of the curves as function of the system height $L_2$. Bottom right inset: rescaled probability density function of the winding angle at different distances $x$; in gray is the Gaussian distribution.

Figure 3

Left-passage probability of spanning perimeters and the prediction (6) for $\kappa ={\kappa }_{\mathrm{LPP}}$. Top right inset: weighted mean-square deviation $Q$ given by (7). Bottom left inset: example of a rigid cluster (blue) and its left and right hulls (black thick lines). The right hull passes to the left of the point located at $z=\rho e^{i\varphi }$ from its origin, marked as a red vector.