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.
In the presence of attraction, the jamming transition of packings of frictionless particles corresponds to the rigidity percolation. When the range of attraction is long, the distribution of the size of rigid clusters, $P(s)$, is continuous and shows a power-law decay. For systems with short-range attractions, however, $P(s)$ appears discontinuous. There is a power-law decay for small cluster sizes, followed by a low probability gap and a peak near the system size. We find that this appearing ``discontinuity'' does not mean that the transition is discontinuous. In fact, it signifies the coexis…
Controlling the connectivity and rigidity of kirigami, i.e. the process of cutting paper to deploy it into an articulated system, is critical in the manifestations of kirigami in art, science and technology, as it provides the resulting metamaterial with a range of mechanical and geometric properties. Here we combine deterministic and stochastic approaches for the control of rigidity in kirigami using the power of $k$ choices, an approach borrowed from the statistical mechanics of explosive percolation transitions. We show that several methods for rigidifying a kirigami system by incrementally…
The discontinuous jump in the bulk modulus $B$ at the jamming transition is a consequence of the formation of a critical contact network of spheres that resists compression. We introduce lattice models with underlying under-coordinated compression resistant spring lattices to which next-nearest-neighbor springs can be added. In these models, the jamming transition emerges as a kind of multicritical point terminating a line of rigidity-percolation transitions. Replacing the under-coordinated lattices with the critical network at jamming yields a faithful description of jamming and its relation …
We conjecture an exact form for an universal ratio of four-point cluster connectivities in the critical two-dimensional $Q$-color Potts model. We also provide analogous results for the limit $Q\rightarrow 1$ that corresponds to percolation where the observable has a logarithmic singularity. Our conjectures are tested against Monte Carlo simulations showing excellent agreement for $Q=1,2,3$.
High-precision Dynamic Monte Carlo Study of Rigidity Percolation
University of Science and Technology of China（中国科学技术大学）；Hefei University of Technology（合肥工业大学）；Hefei National Research Center for Physical Sciences at the Microscales, University of Science and Technology of China（中国科学技术大学微观物质科学国家研究中心）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）
We study the influence of cell-level mechanical heterogeneity in epithelial tissues using a vertex-based model. Heterogeneity in single cell stiffness is introduced as a quenched random variable in the preferred shape index($p_0$) for each cell. We uncovered a crossover scaling for the tissue shear modulus, suggesting that tissue collective rigidity is controlled by a single parameter $f_r$, which accounts for the fraction of rigid cells. Interestingly, the rigidity onset occurs at $f_r=0.21$, far below the contact percolation threshold of rigid cells. Due to the separation of rigidity and con…
We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a "near-loop" when it comes within one lattice spacing of itself. We define a new curve by erasing these near-loops as we traverse the interface. Our Monte Carlo simulations of this model lead us to conclude that the scaling limit of this loop-erased percolation interface is conformally invariant and has fractal dimension 4/3. However, it is not SLE_8/3. We also consider the process in…
The onset of rigidity in interacting liquids, as they undergo a transition to a disordered solid, is associated with a rearrangement of the low-frequency vibrational spectrum. In this letter, we derive scaling forms for the singular dynamical response of disordered viscoelastic networks near both jamming and rigidity percolation. Using effective-medium theory, we extract critical exponents, invariant scaling combinations and analytical formulas for universal scaling functions near these transitions. Our scaling forms describe the behavior in space and time near the various onsets of rigidity, …
University of Science and Technology of China（中国科学技术大学）；Hefei University of Technology（合肥工业大学）；Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China（中国科学技术大学微尺度物质科学国家研究中心）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）
We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a $q$-state Potts model coupled to an $r$-state Potts model in two dimensions. For integer values of $q$ and $r$ the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin-Teller case ($q=r=2$). Since our results extend to continuous values of the number of states, we can access the limit $r\to 1$ corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot…
We study conformal invariance of vorticity clusters in weakly compressible two-dimensional turbulence at low Mach numbers. On the basis of very high resolution direct numerical simulation we demonstrate the scaling invariance of the inverse cascade with scaling close to Kolmogorov prediction. In this range of scales, the statistics of zero-vorticity isolines are found to be compatible with those of critical percolation, thus generalizing the results obtained in incompressible Navier-Stokes turbulence.
