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.