Lower bounds for boundary roughness for droplets in Bernoulli percolation

Details Lower bounds for boundary roughness for droplets in Bernoulli percolation Lower bounds for boundary roughness for droplets in Bernoulli percolation

2002-10-01

arxiv.org/abs/math/0210016v1

Mathematics

Probability

28 pages, 1 figure (.eps file). See also http://math.usc.edu/~alexandr/

Scientific paper

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 $l^{1/3}(\log l)^{-2/3}$. This complements the upper bound of order $l^{1/3}(\log l)^{2/3}$ known 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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