  We use techniques from the theory of electrical networks to give nearly tight
bounds for the transience class of the Abelian sandpile model on the
two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model
is a discrete process on graphs that is intimately related to the phenomenon of
self-organized criticality. In this process, vertices receive grains of sand,
and once the number of grains exceeds their degree, they topple by sending
grains to their neighbors. The transience class of a model is the maximum
number of grains that can be added to the system before it necessarily reaches
its steady-state behavior or, equivalently, a recurrent state. Through a more
refined and global analysis of electrical potentials and random walks, we give
an $O(n^4\log^4{n})$ upper bound and an $\Omega(n^4)$ lower bound for the
transience class of the $n \times n$ grid. Our methods naturally extend to
$n^d$-sized $d$-dimensional grids to give $O(n^{3d - 2}\log^{d+2}{n})$ upper
bounds and $\Omega(n^{3d -2})$ lower bounds.
