| Local limit of packable graphs (2009) | |||||||||
Abstract | |||||||||
| We adapt some planar results into higher dimensions. In particular, it is shown that every unbiased local limit of graphs sphere packed in R^d is d-parabolic (under some additional boundedness assumptions). We then extend parts of the circle packing theory into higher dimensions and derive few geometric corollaries. E.g. every infinite graph ``well'' packed in R^d has either strictly positive isoperimetric (Cheeger) constant or admits arbitrarily large finite sets W with boundary size which satisfies |\partial W| < |W|^{(d-1)/d + o(1)}, were "well" is a local bounded geometry assumption. Some open problems and conjectures are gathered at the end. | |||||||||
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