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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.
The dissection of rectangles into squares
R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte · 1940
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A simple criterion for transience of a reversible Markov chain
T. J. Lyons · 1983
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Geodesics and bounded harmonic functions on infinite planar graphs
S. Northshield · 1991
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Rigidity of infinite (circle) packings
O. Schramm · 1991
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Potential Theory on Infinite Networks, Lecture Notes in Mathematics 1950
P. M. Soardi · 1991
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Fixed points, Koebe uniformization and circle packings
Z.-X. He and O. Schramm · 1993
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Circle boundaries of planar graphs
S. Northshield · 1993
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Hyperbolic and parabolic packings
Z.-X. He and O. Schramm · 1995
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Harmonic functions on planar and almost planar graphs and manifolds, via circle packings
I. Benjamini and O. Schramm · 1996
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Random Walks and Harmonic Functions on Infinite Planar Graphs Using Square Tilings
I. Benjamini and O. Schramm · 1996
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Every graph with a positive cheeger constant contains a tree with a positive cheeger constant
I. Benjamini and O. Schramm · 1997
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R. Diestel · 2005
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J. Oxley · 2011
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The Liouville and the intersection properties are equivalent for planar graphs
I. Benjamini, N. Curien, and A. Georgakopoulos · 2012
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Recurrence of planar graph limits
O. Gurel-Gurevich and A. Nachmias · 2013
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Boundaries of planar graphs, via circle packings
O. Angel, M. T. Barlow, O. Gurel-Gurevich, and A. Nachmias · 2016
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Planar stochastic hyperbolic infinite triangulations
N. Curien · 2016
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The boundary of a square tiling of a graph coincides with the Poisson boundary
A. Georgakopoulos · 2016
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Probability on Trees and Networks
Russell Lyons and Yuval Peres · 2016
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Hyperbolic and parabolic unimodular random maps
O. Angel, T. Hutchcroft, A. Nachmias, and G. Ray · 2018
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A characterization of the locally finite networks admitting non-constant harmonic functions of finite energy
J. Carmesin · 2012
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Coarse geometry and randomness
Itai Benjamini · 2013
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E. Gwynne, J. Miller, and S. Sheffield · 2018
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