2,797 research outputs found
Harmonic Dirichlet Functions on Planar Graphs
Benjamini and Schramm (1996) used circle packing to prove that every
transient, bounded degree planar graph admits non-constant harmonic functions
of finite Dirichlet energy. We refine their result, showing in particular that
for every transient, bounded degree, simple planar triangulation and every
circle packing of in a domain , there is a canonical, explicit bounded
linear isomorphism between the space of harmonic Dirichlet functions on and
the space of harmonic Dirichlet functions on .Microsoft Researc
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
Every planar graph with the Liouville property is amenable
We introduce a strengthening of the notion of transience for planar maps in
order to relax the standard condition of bounded degree appearing in various
results, in particular, the existence of Dirichlet harmonic functions proved by
Benjamini and Schramm. As a corollary we obtain that every planar non-amenable
graph admits Dirichlet harmonic functions
Discrete complex analysis on planar quad-graphs
We develop a linear theory of discrete complex analysis on general
quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon,
Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our
approach based on the medial graph yields more instructive proofs of discrete
analogs of several classical theorems and even new results. We provide discrete
counterparts of fundamental concepts in complex analysis such as holomorphic
functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss
discrete versions of important basic theorems such as Green's identities and
Cauchy's integral formulae. For the first time, we discretize Green's first
identity and Cauchy's integral formula for the derivative of a holomorphic
function. In this paper, we focus on planar quad-graphs, but we would like to
mention that many notions and theorems can be adapted to discrete Riemann
surfaces in a straightforward way.
In the case of planar parallelogram-graphs with bounded interior angles and
bounded ratio of side lengths, we construct a discrete Green's function and
discrete Cauchy's kernels with asymptotics comparable to the smooth case.
Further restricting to the integer lattice of a two-dimensional skew coordinate
system yields appropriate discrete Cauchy's integral formulae for higher order
derivatives.Comment: 49 pages, 8 figure
Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem
Consider a planar, bounded, -connected region , and let
\bord\Omega be its boundary. Let be a cellular decomposition of
\Omega\cup\bord\Omega, where each 2-cell is either a triangle or a
quadrilateral. From these data and a conductance function we construct a
canonical pair where is a genus singular flat surface tiled
by rectangles and is an energy preserving mapping from
onto .Comment: 27 pages, 11 figures; v2 - revised definition (now denoted by the
flux-gradient metric (1.9)) in section 1 and minor modifications of proofs;
corrected typo
The Liouville and the intersection properties are equivalent for planar graphs
It is shown that if a planar graph admits no non-constant bounded harmonic
functions then the trajectories of two independent simple random walks
intersect almost surely.Comment: 4 pages, 1 figur
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