The Annals of Probability

Divergence of Shape Fluctuations in Two Dimensions

Charles M. Newman and Marcelo S. T. Piza

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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 $d$.

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Ann. Probab., Volume 23, Number 3 (1995), 977-1005.

First available in Project Euclid: 19 April 2007

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C24: Interface problems; diffusion-limited aggregation 82B24: Interface problems; diffusion-limited aggregation 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

First-passage percolation shape fluctuations roughness exponents Ising model polymers directed polymers random environment stochastic growth


Newman, Charles M.; Piza, Marcelo S. T. Divergence of Shape Fluctuations in Two Dimensions. Ann. Probab. 23 (1995), no. 3, 977--1005. doi:10.1214/aop/1176988171.

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  • See Correction: Charles M. Newman, Marcelo S. T. Piza. Correction: Divergence of Shape Fluctuations in Two Dimensions. Ann. Probab., Volume 23, Number 4 (1995), 2057--2057.