  We provide a definition of a new critical exponent $\beta$ that has the
interpretation of a type of local walk dimension, and may be defined on any
compact metric space. We then specialize to the case of random walks that jump
uniformly in metric balls with respect to a given Borel measure of full
support. We use the local exponent $\beta$ as a local time scaling exponent to
re-normalize the time scale and produce approximating continuous time walks. We
show a Faber-Krahn type inequality $\lambda_{1,r}(B)\geq
\frac{c}{R^{\beta(x_0)}},$ where $c$ is a constant independent of $r$ and $x_0$
and where $\lambda_{1,r}(B)$ is the bottom of the spectrum of the generator for
the re-normalized continuous time walk at stage $r$ killed outside of
$B=B_R(x_0).$ In addition, we examine the local Hausdorff dimension $\alpha.$
We show that any variable Ahlfors $Q$-regular measure is strongly equivalent to
the local Hausdorff measure and that $Q=\alpha.$ We also provide new examples
of variable dimensional spaces, including a variable dimensional Sierpinski
carpet.
