Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Boundaries of planar graphs, via circle packings   We provide a geometric representation of the Poisson and Martin boundaries of
a transient, bounded degree triangulation of the plane in terms of its circle
packing in the unit disc. (This packing is unique up to M\"obius
transformations.) More precisely, we show that any bounded harmonic function on
the graph is the harmonic extension of some measurable function on the boundary
of the disk, and that the space of extremal positive harmonic functions, that
is, the Martin boundary, is homeomorphic to the unit circle. All our results
hold more generally for any "good"-embedding of planar graphs, that is, an
embedding in the unit disc with straight lines such that angles are bounded
away from $0$ and $\pi$ uniformly, and lengths of adjacent edges are
comparable. Furthermore, we show that in a good embedding of a planar graph the
probability that a random walk exits a disc through a sufficiently wide arc is
at least a constant, and that Brownian motion on such graphs takes time of
order $r^2$ to exit a disc of radius $r$. These answer a question recently
posed by Chelkak (2014).

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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: eess