Thresholds for Pebbling on Grids

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.