  Given a connected graph $G$ and a configuration of $t$ pebbles on the
vertices of G, a $q$-pebbling step consists of removing $q$ pebbles from a
vertex, and adding a single pebble to one of its neighbors. Given a vector
$\bf{q}=(q_1,\ldots,q_d)$, $\bf{q}$-pebbling consists of allowing
$q_i$-pebbling in coordinate $i$. A distribution of pebbles is called solvable
if it is possible to transfer at least one pebble to any specified vertex of
$G$ via a finite sequence of pebbling steps.
  In this paper, we determine the weak threshold for $\bf{q}$-pebbling on the
sequence of grids $[n]^d$ for fixed $d$ and $\bf{q}$, as $n\to\infty$. Further,
we determine the strong threshold for $q$-pebbling on the sequence of paths of
increasing length. A fundamental tool in these proofs is a new notion of
centrality, and a sufficient condition for solvability based on the well used
pebbling weight functions; we believe this weight lemma to be the first result
of its kind, and may be of independent interest.
  These theorems improve recent results of Czygrinow and Hurlbert, and Godbole,
Jablonski, Salzman, and Wierman. They are the generalizations to the random
setting of much earlier results of Chung.
  In addition, we give a short counterexample showing that the threshold
version of a well known conjecture of Graham does not hold. This uses a result
for hypercubes due to Czygrinow and Wagner.
