We study a new geometric bootstrap percolation model,
We study site percolation models on planar lattices including the [
- NSF-PAR ID:
- 10148983
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 57
- Issue:
- 2
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 474-525
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract line percolation , on thed ‐dimensional integer grid. In line percolation with infection parameter r , infection spreads from a subsetof initially infected lattice points as follows: if there exists an axis‐parallel line L withr or more infected lattice points on it, then every lattice point ofon L gets infected, and we repeat this until the infection can no longer spread. The elements of the setA are usually chosen independently, with some densityp , and the main question is to determine, the density at which percolation (infection of the entire grid) becomes likely. In this paper, we determine up to a multiplicative factor of and up to a multiplicative constant as for every fixed . We also determine the size of the minimal percolating sets in all dimensions and for all values of the infection parameter. -
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We consider instances of long‐range percolation on
and , where points at distance r get connected by an edge with probability proportional tor −s , fors ∈ (d ,2d ), and study the asymptotic of the graph‐theoretical (a.k.a. chemical) distanceD (x ,y ) betweenx andy in the limit as |x −y |→∞ . For the model onwe show that, in probability as | x |→∞ , the distanceD (0,x ) is squeezed between two positive multiples of, where for γ : =s /(2d ). For the model onwe show that D (0,xr ) is, in probability asr →∞ for any nonzero, asymptotic to for φ a positive, continuous (deterministic) function obeyingφ (r γ ) =φ (r ) for allr > 1. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly‐exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain. -
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Abstract The mineral apatite, Ca5(PO4)3(F,Cl,OH), is a ubiquitous accessory mineral, with its volatile content and isotopic compositions used to interpret the evolution of H2O on planetary bodies. During hypervelocity impact, extreme pressures shock target rocks resulting in deformation of minerals; however, relatively few microstructural studies of apatite have been undertaken. Given its widespread distribution in the solar system, it is important to understand how apatite responds to progressive shock metamorphism. Here, we present detailed microstructural analyses of shock deformation in ~560 apatite grains throughout ~550 m of shocked granitoid rock from the peak ring of the Chicxulub impact structure, Mexico. A combination of high‐resolution backscattered electron (BSE) imaging, electron backscatter diffraction mapping, transmission Kikuchi diffraction mapping, and transmission electron microscopy is used to characterize deformation within apatite grains. Systematic, crystallographically controlled deformation bands are present within apatite, consistent with tilt boundaries that contain the <
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Purpose Recent observations of several preferred orientations of diffusion in deep white matter may indicate either (a) that axons in different directions are independently bundled in thick sheets and function noninteractively, or more interestingly, (b) that the axons are closely interwoven and would exhibit branching and sharp turns. This study aims to investigate whether the dependence of dMRI Q‐ball signal on the interpulse time
can decode the smaller‐than‐voxel‐size brain structure, in particular, to distinguish scenarios (a) and (b). Methods High‐resolution Q‐ball images of a healthy brain taken with
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