An improved upper bound for the pebbling threshold of the n-path

Given a configuration of t indistinguishable pebbles on the n vertices of a graph G, we say that a vertex v can be reached if a pebble can be placed on it in a finite number of “moves”. G is said to be pebbleable if all its vertices can be thus reached. Now given the n-path Pn how large (resp. small) must t be so as to be able to pebble the path almost surely (resp. almost never)? It was known that the threshold th(Pn) for pebbling the path satisfies n2clg n⩽th(Pn)⩽n22lg n, where lg=log2 and c<1/2 is arbitrary. We improve the upper bound for the threshold function to th(Pn)⩽n2dlg n, where d>1 is arbitrary.