DIVERGENCE OF SHAPE FLUCTUATIONS IN 2 DIMENSIONS

We consider stochastic growth models, such as standard first-passage p ercolation on Z(d), where to leading order there is a linearly growing deterministic shape. Under natural hypotheses, we prove that for d = 2, the shape fluctuations grow at least logarithmically in all directi ons. Although this bound is far from the expected power law behavior w ith exponent chi = 1/3, it does prove divergence. With additional hypo theses, we obtain inequalities involving chi and the related exponent xi (which is expected to equal 2/3 for d = 2). Combining these inequal ities with previously known results, we obtain for standard first-pass age percolation the bounds chi greater than or equal to 1/8 for d = 2 and xi greater than or equal to 3/4 for all d.