Lower bounds for boundary roughness for droplets in Bernoulli percolation |
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Authors: | Hasan B?Uzun Email author" target="_blank">Kenneth S?AlexanderEmail author |
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Institution: | (1) ALEKS Corporation, 400 N. Tustin Avenue, Suite 300, Santa Ana, CA 92705, USA;(2) Department of Mathematics, University of Southern California, DRB 155, Los Angeles, CA 90089-1113, USA |
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Abstract: | We consider boundary roughness for the ``droplet' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l2, for large l. The maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull; we show that for large l this maximum is at least of order l1/3(logl)–2/3. This complements the upper bound of order l1/3(logl)–2/3 proved in Al3] for the average local roughness. The exponent 1/3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions.
The research of the first author was supported by NSF grant DMS-9802368. The research of the second author was supported by NSF grants DMS-9802368 and DMS-0103790.Mathematics Subject Classification (2000): Primary 60K35; Secondary 82B20, 82B43 |
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Keywords: | Droplet Interface Local roughness |
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