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 l 2, 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 l 1/3(logl)−2/3. This complements the upper bound of order l 1/3(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.
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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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Uzun, H., Alexander, K. Lower bounds for boundary roughness for droplets in Bernoulli percolation. Probab. Theory Relat. Fields 127, 62–88 (2003). https://doi.org/10.1007/s00440-003-0276-0
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DOI: https://doi.org/10.1007/s00440-003-0276-0