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Given a finite simple triangulation, we estimate the sizes of circles in its circle packing in terms of Cannon's vertex extremal length.
Kontaktprobleme der konformen abbildung
P. Koebe · 1936
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Potential theory on a rhombic lattice
R. J. Duffin · 1968
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The combinatorial Riemann mapping theorem
J. W. Cannon · 1994
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Hyperbolic and parabolic packings
Z.-X. He and O. Schramm · 1995
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Discrete solutions of Dirichlet problems, finite volumes, and circle packings
T. Dubejko · 1999
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Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits
S. Smirnov · 2001
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Introduction to circle packing
K. Stephenson · 2005
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Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
S. Smirnov · 2010
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Discrete complex analysis and probability
S. Smirnov · 2010
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Discrete complex analysis on isoradial graphs
D. Chelkak and S. Smirnov · 2011
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Liouville quantum gravity and KPZ
B. Duplantier and S. Sheffield · 2011
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Liouville quantum gravity as a mating of trees
B. Duplantier, J. Miller, and S. Sheffield
Cited in the paper.
The Dirichlet problem for orthodiagonal maps
Universality in the 2D Ising model and conformal invariance of fermionic observables
D. Chelkak and S. Smirnov · 2012
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Quantum gravity and the KPZ formula [after Duplantier-Sheffield]
C. Garban · 2012
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The boundary value problem for discrete analytic functions
M. Skopenkov · 2013
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Random geometry on the sphere
J.-F. Le Gall · 2014
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Graph theory
R. Diestel · 2017
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O. Gurel-Gurevich, D. C. Jerison, and A. Nachmias
Cited in the paper.
Random walk on random planar maps: spectral dimension, resistance, and displacement
E. Gwynne and J. Miller
Cited in the paper.
Harmonic functions on mated-CRT maps
E. Gwynne, J. Miller, and S. Sheffield
Cited in the paper.
The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity
E. Gwynne, J. Miller, and S. Sheffield
Cited in the paper.
Planar maps, random walks and circle packing
A. Nachmias
Cited in the paper.
Discrete analytic functions on non-uniform lattices without global geometric control
B. M. Werness
Cited in the paper.