On pebbling threshold functions for graph sequences

Given a connected graph G, and a distribution of / pebbles to the vertices of G, a pebbling step consists of removing two pebbles from a vertex v and placing one pebble on a neighbor of v. For a particular vertex r, the distribution is r-solvable if it is possible to place a pebble on r after a finite number of pebbling steps. The distribution is solvable if it is r-solvable for every r. The pebbling number of G is the least number /, so that every distribution of t pebbles is solvable. In this paper we are not concerned with such an absolute guarantee but rather an almost sure guarantee. A threshold function for a sequence of graphs 'S = (Gi,G2,...,G,...), where G has n vertices, is any function ta(n) such that almost all distributions of / pebbles are solvable when t>t₀, and such that almost none are solvable when t<$to. We give bounds on pebbling threshold functions for the sequences of cliques, stars, wheels, cubes, cycles and paths.