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Lower bounds for boundary roughness for droplets in Bernoulli percolation
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  • Published: 18 June 2003

Lower bounds for boundary roughness for droplets in Bernoulli percolation

  • Hasan B. Uzun1 &
  • Kenneth S. Alexander2 

Probability Theory and Related Fields volume 127, pages 62–88 (2003)Cite this article

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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 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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Author information

Authors and Affiliations

  1. ALEKS Corporation, 400 N. Tustin Avenue, Suite 300, Santa Ana, CA, 92705, USA

    Hasan B. Uzun

  2. Department of Mathematics, University of Southern California, DRB 155, Los Angeles, CA, 90089-1113, USA

    Kenneth S. Alexander

Authors
  1. Hasan B. Uzun
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  2. Kenneth S. Alexander
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Corresponding author

Correspondence to Kenneth S. Alexander.

Additional information

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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  • Received: 26 November 2002

  • Published: 18 June 2003

  • Issue Date: September 2003

  • DOI: https://doi.org/10.1007/s00440-003-0276-0

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Keywords

  • Droplet
  • Interface
  • Local roughness
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