Divergence of Shape Fluctuations in Two Dimensions

Newman, Charles M. ; Piza, Marcelo S. T.

Ann. Probab., Tome 23 (1995) no. 3, p. 977-1005 / Harvested from Project Euclid

Résumé

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 expected power law behavior with exponent $\chi = 1/3$, it does prove divergence. With additional hypotheses, we obtain inequalities involving $\chi$ and the related exponent $\xi$ (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 $\chi \geq 1/8$ for $d = 2$ and $\xi \leq 3/4$ for all $d$.

Détail

Publié le : 1995-07-14
Classification:  First-passage percolation,  shape fluctuations,  roughness exponents,  Ising model,  polymers,  directed polymers,  random environment,  stochastic growth,  60K35,  82C24,  82B24,  82B44