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Correction-to-scaling exponent for two-dimensional percolation.

Robert M Ziff1

  • 1Center for the Study of Complex Systems and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA. rziff@umich.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 17, 2011
PubMed
Summary
This summary is machine-generated.

New bounds for correction-to-scaling exponents in 2D percolation were established. These upper bounds align with experimental data and theoretical predictions, suggesting their exactness and applicability to site percolation models.

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Area of Science:

  • Statistical physics
  • Complex systems

Background:

  • Percolation theory studies the connectivity of random networks.
  • Correction-to-scaling exponents describe deviations from critical behavior in physical systems.

Purpose of the Study:

  • To establish rigorous upper bounds for correction-to-scaling exponents in two-dimensional percolation.
  • To verify the consistency of these bounds with existing experimental and theoretical results.

Main Methods:

  • Utilizing Cardy's result for crossing probability on an annulus.
  • Applying theoretical analysis to derive exponent bounds.

Main Results:

  • Established upper bounds: Ω ≤ 72/91, ω = DΩ ≤ 3/2, and Δ₁ = νω ≤ 2.
  • Demonstrated consistency of these bounds with site percolation measurements on square and triangular lattices.
  • Showed agreement with new bond percolation measurements and theoretical exponents for hulls.

Conclusions:

  • The derived upper bounds for correction-to-scaling exponents are likely exact.
  • A correction-to-scaling form is applicable to site percolation.
  • Findings support recent theoretical proposals for hull exponents.