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March, 2004 Which values of the volume growth and escape time exponent are possible for a graph?
Martin T. Barlow
Rev. Mat. Iberoamericana 20(1): 1-31 (March, 2004).

Abstract

Let $\Gamma=(G,E)$ be an infinite weighted graph which is Ahlfors $\alpha$-regular, so that there exists a constant $c$ such that $c^{-1} r^\alpha\le V(x,r)\le c r^\alpha$, where $V(x,r)$ is the volume of the ball centre $x$ and radius $r$. Define the escape time $T(x,r)$ to be the mean exit time of a simple random walk on $\Gamma$ starting at $x$ from the ball centre $x$ and radius $r$. We say $\Gamma$ has escape time exponent $\beta>0$ if there exists a constant $c$ such that $c^{-1} r^\beta \le T(x,r) \le c r^\beta$ for $r\ge 1$. Well known estimates for random walks on graphs imply that $\alpha\ge 1$ and $2 \le \beta \le 1+\alpha$. We show that these are the only constraints, by constructing for each $\alpha_0$, $\beta_0$ satisfying the inequalities above a graph $\widetilde{\Gamma}$ which is Ahlfors $\alpha_0$-regular and has escape time exponent $\beta_0$. In addition we can make $\widetilde{\Gamma}$ sufficiently uniform so that it satisfies an elliptic Harnack inequality.

Citation

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Martin T. Barlow. "Which values of the volume growth and escape time exponent are possible for a graph?." Rev. Mat. Iberoamericana 20 (1) 1 - 31, March, 2004.

Information

Published: March, 2004
First available in Project Euclid: 2 April 2004

zbMATH: 1051.60071
MathSciNet: MR2076770

Subjects:
Primary: 60J05
Secondary: 58J65 , 82C41

Keywords: Anomalous diffusion , graph , Random walk , volume growth

Rights: Copyright © 2004 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.20 • No. 1 • March, 2004
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