On limits of Graphs Sphere Packed in Euclidean Space and Applications

The core of this note is the observation that links between circle packings of graphs and potential theory developed in \cite{BeSc01} and \cite{HS} can be extended to higher dimensions. In particular, it is shown that every limit of finite graphs sphere packed in $\R^d$ with a uniformly-chosen root is $d$-parabolic. We then derive few geometric corollaries. E.g.\,every infinite graph packed in $\R^{d}$ has either strictly positive isoperimetric Cheeger constant or admits arbitrarily large finite sets $W$ with boundary size which satisfies $|\partial W| \leq |W|^{\frac{d-1}{d}+o(1)}$. Some open problems and conjectures are gathered at the end

Lack of Sphere Packing of Graphs via Non-Linear Potential Theory

It is shown that there is no quasi-sphere packing of the lattice grid Z^{d+1} or a co-compact hyperbolic lattice of H^{d+1} or the 3-regular tree \times Z, in R^d, for all d. A similar result is proved for some other graphs too. Rather than using a direct geometrical approach, the main tools we are using are from non-linear potential theory.Comment: 10 page

The Tensor Track, III

We provide an informal up-to-date review of the tensor track approach to quantum gravity. In a long introduction we describe in simple terms the motivations for this approach. Then the many recent advances are summarized, with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion, Osterwalder-Schrader positivity...) which, while important for the tensor track program, are not detailed in the usual quantum gravity literature. We list open questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure

Planar maps, circle patterns and 2d gravity

Via circle pattern techniques, random planar triangulations (with angle variables) are mapped onto Delaunay triangulations in the complex plane. The uniform measure on triangulations is mapped onto a conformally invariant spatial point process. We show that this measure can be expressed as: (1) a sum over 3-spanning-trees partitions of the edges of the Delaunay triangulations; (2) the volume form of a K\"ahler metric over the space of Delaunay triangulations, whose prepotential has a simple formulation in term of ideal tessellations of the 3d hyperbolic space; (3) a discretized version (involving finite difference complex derivative operators) of Polyakov's conformal Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes, thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17 figure