Correlated percolation and the correlated resistor network

We present some exact results on percolation properties of the Ising model, when the range of the percolating bonds is larger than nearest-neighbors. We show that for a percolation range to next-nearest neighbors the percolation threshold Tp is still equal to the Ising critical temperature Tc, and present the phase diagram for this type of percolation. In addition, we present Monte Carlo calculations of the finite size behavior of the correlated resistor network defined on the Ising model. The thermal exponent t of the conductivity that follows from it is found to be t = 0.2000 +- 0.0007. We observe no corrections to scaling in its finite size behavior.Comment: 16 pages, REVTeX, 6 figures include

Statistics of correlated percolation in a bacterial community

Signal propagation over long distances is a ubiquitous feature of multicellular communities, but cell-to-cell variability can cause propagation to be highly heterogeneous. Simple models of signal propagation in heterogenous media, such as percolation theory, can potentially provide a quantitative understanding of these processes, but it is unclear whether these simple models properly capture the complexities of multicellular systems. We recently discovered that in biofilms of the bacterium Bacillus subtilis, the propagation of an electrical signal is statistically consistent with percolation theory, and yet it is reasonable to suspect that key features of this system go beyond the simple assumptions of basic percolation theory. Indeed, we find here that the probability for a cell to signal is not independent from other cells as assumed in percolation theory, but instead is correlated with its nearby neighbors. We develop a mechanistic model, in which correlated signaling emerges from cell division, phenotypic inheritance, and cell displacement, that reproduces the experimentally observed correlations. We find that the correlations do not significantly affect the spatial statistics, which we rationalize using a renormalization argument. Moreover, the fraction of signaling cells is not constant in space, as assumed in percolation theory, but instead varies within and across biofilms. We find that this feature lowers the fraction of signaling cells at which one observes the characteristic power-law statistics of cluster sizes, consistent with our experimental results. We validate the model using a mutant biofilm whose signaling probability decays along the propagation direction. Our results reveal key statistical features of a correlated signaling process in a multicellular community. More broadly, our results identify extensions to percolation theory that do or do not alter its predictions and may be more appropriate for biological systems.P50 GM085764 - NIGMS NIH HHS; Howard Hughes Medical Institute; R01 GM121888 - NIGMS NIH HHSPublished versio

Flow correlated percolation during vascular network formation in tumors

A theoretical model based on the molecular interactions between a growing tumor and a dynamically evolving blood vessel network describes the transformation of the regular vasculature in normal tissues into a highly inhomogeneous tumor specific capillary network. The emerging morphology, characterized by the compartmentalization of the tumor into several regions differing in vessel density, diameter and necrosis, is in accordance with experimental data for human melanoma. Vessel collapse due to a combination of severely reduced blood flow and solid stress exerted by the tumor, leads to a correlated percolation process that is driven towards criticality by the mechanism of hydrodynamic vessel stabilization.Comment: 4 pages, 3 figures (higher resolution at http://www.uni-saarland.de/fak7/rieger/HOMEPAGE/flow.eps

Flat-Band Ferromagnetism as a Pauli-Correlated Percolation Problem

We investigate the location and nature of the para-ferro transition of interacting electrons in dispersionless bands using the example of the Hubbard model on the Tasaki lattice. This case can be analyzed as a geometric site-percolation problem where different configurations appear with nontrivial weights. We provide a complete exact solution for the 1D case and develop a numerical algorithm for the 2D case. In two dimensions the paramagnetic phase persists beyond the uncorrelated percolation point, and the grand-canonical transition is via a first-order jump to an unsaturated ferromagnetic phase.Comment: 6 pages, 5 figure

Origin of the Universal Roughness Exponent of Brittle Fracture Surfaces: Correlated Percolation in the Damage Zone

We suggest that the observed large-scale universal roughness of brittle fracture surfaces is due to the fracture process being a correlated percolation process in a self-generated quadratic damage gradient. We use the quasi-static two-dimensional fuse model as a paradigm of a fracture model. We measure for this model, that exhibits a correlated percolation process, the correlation length exponent nu approximately equal to 1.35 and conjecture it to be equal to that of uncorrelated percolation, 4/3. We then show that the roughness exponent in the fuse model is zeta = 2 nu/(1+2 nu)= 8/11. This is in accordance with the numerical value zeta=0.75. As for three-dimensional brittle fractures, a mean-field theory gives nu=2, leading to zeta=4/5 in full accordance with the universally observed value zeta =0.80.Comment: 4 pages RevTeX

Roughness of Interfacial Crack Front: Correlated Percolation in the Damage Zone

We show that the roughness exponent zeta of an in-plane crack front slowly propagating along a heterogeneous interface embeded in a elastic body, is in full agreement with a correlated percolation problem in a linear gradient. We obtain zeta=nu/(1+nu) where nu is the correlation length critical exponent. We develop an elastic brittle model based on both the 3D Green function in an elastic half-space and a discrete interface of brittle fibers and find numerically that nu=1.5, We conjecture it to be 3/2. This yields zeta=3/5. We also obtain by direct numerical simulations zeta=0.6 in excellent agreement with our prediction. This modelling is for the first time in close agreement with experimental observations.Comment: 4 pages RevTeX