Quasisymmetric uniformization and heat kernel estimates

Murugan, Mathav

doi:10.1090/tran/7713

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.

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