Lower bounds for boundary roughness for droplets in Bernoulli percolation

2002/10/01 by Hasan Uzun, Hasan B. Uzun, Uzun, Hasan B. +2
Computer Science · Mathematics · Physics and Astronomy · #60K35 #82B20 #82B43 #Bayesian Methods and Mixture Models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60K35 #msc:82B20 #msc:82B43

paper · pdf · doi:10.48550/arxiv.math/0210016

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

arxiv created 2002/10/01 · openalex publication_date 2002/10/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

Related

• Phase separation in random cluster models I: uniform upper bounds on local deviation
• Phase separation in random cluster models II: the droplet at equilibrium, and local deviation lower bounds
• The size of the boundary in first-passage percolation
• There is no isolated interface edge in very supercritical percolation
• The box-crossing property for critical two-dimensional oriented percolation