Open Access
Translator Disclaimer
July, 1995 Divergence of Shape Fluctuations in Two Dimensions
Charles M. Newman, Marcelo S. T. Piza
Ann. Probab. 23(3): 977-1005 (July, 1995). DOI: 10.1214/aop/1176988171


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 $d$.


Download Citation

Charles M. Newman. Marcelo S. T. Piza. "Divergence of Shape Fluctuations in Two Dimensions." Ann. Probab. 23 (3) 977 - 1005, July, 1995.


Published: July, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0835.60087
MathSciNet: MR1349159
Digital Object Identifier: 10.1214/aop/1176988171

Primary: 60K35
Secondary: 82B24 , 82B44 , 82C24

Keywords: Directed polymers , First-passage percolation , Ising model , Polymers , random environment , roughness exponents , shape fluctuations , Stochastic growth

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 3 • July, 1995
Back to Top