Divergence of Shape Fluctuations in Two Dimensions
Divergence of Shape Fluctuations in Two Dimensions
We consider stochastic growth models, such as standard first-passage percolation on $\mathbb{Z}^d$, 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 directions. Although this bound is far from the …