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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quasisymmetric uniformization and heat kernel estimates

Author: Mathav Murugan
Journal: Trans. Amer. Math. Soc. 372 (2019), 4177-4209
MSC (2010): Primary 60J45, 51F99
Published electronically: April 25, 2019
MathSciNet review: 4009428
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Abstract: We show that the circle packing embedding in $\mathbb {R}^2$ of a one-ended, planar triangulation with polynomial growth is quasisymmetric if and only if the simple random walk on the graph satisfies sub-Gaussian heat kernel estimate with spectral dimension two. Our main results provide a new family of graphs and fractals that satisfy sub-Gaussian estimates and Harnack inequalities.

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Additional Information

Mathav Murugan
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
MR Author ID: 864378

Keywords: Quasisymmetry, uniformization, circle packing, sub-Gaussian estimate, Harnack inequality.
Received by editor(s): March 29, 2018
Received by editor(s) in revised form: August 13, 2018
Published electronically: April 25, 2019
Additional Notes: The author’s research was partially supported by NSERC (Canada) and the Pacific Institute for the Mathematical Sciences
Dedicated: Dedicated to Professor Laurent Saloff-Coste on the occasion of his 60th birthday
Article copyright: © Copyright 2019 American Mathematical Society