We study the following game on a finite graph G=(V,E). At the start,
each edge is assigned an integer ne≥0, n=∑e∈Ene. In
round t, 1≤t≤n, a uniformly random vertex v∈V is chosen and
one of the edges f incident with v is selected by the player. The value
assigned to f is then decreased by 1. The player wins, if the configuration
(0,…,0) is reached; in other words, the edge values never go negative.
Our main result is that there is a phase transition: as n→∞, the
probability that the player wins approaches a constant cG>0 when (ne/n:e∈E) converges to a point in the interior of a certain convex set
RG, and goes to 0 exponentially when (ne/n:e∈E) is
bounded away from RG. We also obtain upper bounds in the
near-critical region, that is when (ne/n:e∈E) lies close to ∂RG. We supply quantitative error bounds in our arguments.Comment: 28 pages, 2 eps figures. Some mistakes have been corrected, and the
introduction has been re-written. Minor corrections throughou