Before revision:   We find that  load balance in a system of n nodes improves multiplicatively with d as long as d = o\left(\log(n)\right), and improves exponentially as soon as d = \Theta\left(\log(n)\right).
After revision:  In our model, the load offered for the objects is sampled uniformly at random from all the load vectors with a fixed cumulative value. We find that load balance in a system of n nodes improves multiplicatively with d as long as d = o\left(\log(n)\right), and improves exponentially as soon as d = \Theta\left(\log(n)\right).