JournalsrmiVol. 20, No. 1pp. 1–31

Which values of the volume growth and escape time exponent are possible for a graph?

  • Martin T. Barlow

    The University of British Columbia, Vancouver, Canada
Which values of the volume growth and escape time exponent are possible for a graph? cover
Download PDF

Abstract

Let Γ=(G,E)\Gamma=(G,E) be an infinite weighted graph which is Ahlfors α\alpha-regular, so that there exists a constant cc such that c1rαV(x,r)crαc^{-1} r^\alpha\le V(x,r)\le c r^\alpha, where V(x,r)V(x,r) is the volume of the ball centre xx and radius rr. Define the escape time T(x,r)T(x,r) to be the mean exit time of a simple random walk on Γ\Gamma starting at xx from the ball centre xx and radius rr. We say Γ\Gamma has escape time exponent β>0\beta>0 if there exists a constant cc such that c1rβT(x,r)crβc^{-1} r^\beta \le T(x,r) \le c r^\beta for r1r\ge 1. Well known estimates for random walks on graphs imply that α1\alpha\ge 1 and 2β1+α2 \le \beta \le 1+\alpha. We show that these are the only constraints, by constructing for each α0\alpha_0, β0\beta_0 satisfying the inequalities above a graph Γ~\widetilde{\Gamma} which is Ahlfors α0\alpha_0-regular and has escape time exponent β0\beta_0. In addition we can make Γ~\widetilde{\Gamma} sufficiently uniform so that it satisfies an elliptic Harnack inequality.

Cite this article

Martin T. Barlow, Which values of the volume growth and escape time exponent are possible for a graph?. Rev. Mat. Iberoam. 20 (2004), no. 1, pp. 1–31

DOI 10.4171/RMI/378