Quasisymmetric uniformization and heat kernel estimates
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- Trans. Amer. Math. Soc. 372 (2019), 4177-4209 Request permission
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.References
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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
- Email: mathav@math.ubc.ca
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 4177-4209
- MSC (2010): Primary 60J45, 51F99
- DOI: https://doi.org/10.1090/tran/7713
- MathSciNet review: 4009428
Dedicated: Dedicated to Professor Laurent Saloff-Coste on the occasion of his 60th birthday