Divergence of Shape Fluctuations in Two Dimensions

We consider stochastic growth models, such as standard first-passage percolation on Zᵈ, 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 directions. Although this bound is far from the expected power law behavior with exponent χ = 1/3, it does prove divergence. With additional hypotheses, we obtain inequalities involving χ and the related exponent ξ (which is expected to equal 2/3 for $d = 2$). Combining these inequalities with previously known results, we obtain for standard first-passage percolation the bounds χ ≥ 1/8 for $d = 2$ and ξ ≤ 3/4 for all d.