Related papers: Some criteria for circle packing types and combina…
Let $\mathbb{R}_{+}^{n+1}$ \ be the half-space model of the hyperbolic space $\mathbb{H}^{n+1}.$ It is proved that if $\Gamma\subset\left\{ x_{n+1}=0\right\} \subset\partial_{\infty}\mathbb{H}^{n+1}$ is a bounded $C^{0}$ Euclidean graph…
We consider random walks on finite vertex-transitive graphs $\Gamma$ of bounded degree. We find a simple geometric condition which characterises the cover time fluctuations: the suitably normalised cover time converges to a standard Gumbel…
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…
We study the implications of the simplicity constraint in the spincube model of quantum gravity. By relating the edge-lengths to the integer areas of triangles, the simplicity constraint imposes very strong restrictions between them,…
Given a discrete subgroup $\Gamma$ of finite co-volume of $\mathrm{PGL}(2,\mathbb{R})$, we define and study parabolic vector bundles on the quotient $\Sigma$ of the (extended) hyperbolic plane by $\Gamma$. If $\Gamma$ contains an…
A foundation is laid for a theory of combinatorial groupoids, allowing us to use concepts like ``holonomy'', ``parallel transport'', ``bundles'', ``combinatorial curvature'' etc. in the context of simplicial (polyhedral) complexes, posets,…
Gems are a particular type of edge-colored graphs, dual to colored triangulations, which represent compact PL-manifolds of arbitrary dimension, both in the closed and boundary case. In the present paper, gem theory is used to approach…
The Andreev-Thurston Circle Packing Theorem is generalized to packings of convex bodies in planar simply connected domains. This turns out to be a useful tool for constructing conformal and quasiconformal mappings with interesting geometric…
We consider the disordered monomer-dimer model on general finite graphs with bounded degrees. Under the finite fourth moment assumption on the weight distributions, we prove a Gaussian central limit theorem for the free energy of the…
An \emph{$H$-packing} in a graph $G$ is a collection of pairwise vertex-disjoint copies of $H$ in $G$. We prove that for every $c > 0$ and every bipartite graph $H$, any $\lfloor cn \rfloor$-regular graph $G$ admits an $H$-packing that…
We analyze the concept of causality for the conductivity of graphene described by the Dirac model. It is recalled that the condition of causality leads to the analyticity of conductivity in the upper half-plane of complex frequencies and to…
We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of…
The unions of directed graphs are the simplest examples of pushouts of directed graphs. The conditions under which they contravariantly induce surjective gauge-equivariant pullbacks of graph C*-algebras have been well studied and vastly…
Given a polygon $P$ in the triangular grid, we obtain a permutation $\pi_P$ via a natural billiards system in which beams of light bounce around inside of $P$. The different cycles in $\pi_P$ correspond to the different trajectories of…
The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.
We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number…
We prove a discrete Gauss-Bonnet-Chern theorem which states where summing the curvature over all vertices of a finite graph G=(V,E) gives the Euler characteristic of G.
Let G denote a graph and let K be a subset of vertices that are a set of target vertices of G. The K-terminal reliability of G is defined as the probability that all target vertices in K are connected, considering the possible failures of…
We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model…
Let $\Gamma$ be a connected bridgeless metric graph, and fix a point $v$ of $\Gamma$. We define combinatorial iterated integrals on $\Gamma$ along closed paths at $v$, a unipotent generalization of the usual cycle pairing and the…