Area distribution and scaling function for punctured polygons

Richard C, Jensen I, Guttmann A (2008)

Publication Type: Journal article, Online publication

Publication year: 2008

Journal

Electronic Journal of Combinatorics The Electronic Journal of Combinatorics

Publisher: The Electronic Journal of Combinatorics

Book Volume: 15

Pages Range: Research Paper 53,50

Journal Issue: 1

URI: http://www.combinatorics.org/Volume_15/Abstracts/v15i1r53.html

Abstract

Punctured polygons are polygons with internal holes which are also polygons. The external and internal polygons are of the same type, and they are mutually as well as self-avoiding. Based on an assumption about the limiting area distribution for unpunctured polygons, we rigorously analyse the effect of a finite number of punctures on the limiting area distribution in a uniform ensemble, where punctured polygons with equal perimeter have the same probability of occurrence. Our analysis leads to conjectures about the scaling behaviour of the models. We also analyse exact enumeration data. For staircase polygons with punctures of fixed size, this yields explicit expressions for the generating functions of the first few area moments. For staircase polygons with punctures of arbitrary size, a careful numerical analysis yields very accurate estimates for the area moments. Interestingly, we find that the leading correction term for each area moment is proportional to the corresponding area moment with one less puncture. We finally analyse corresponding quantities for punctured self-avoiding polygons and find agreement with the conjectured formulas to at least 3-4 significant digits.

Authors with CRIS profile

Christoph Richard Naturwissenschaftliche Fakultät

Involved external institutions

The University of Melbourne AU Australia (AU)

How to cite

APA:

Richard, C., Jensen, I., & Guttmann, A. (2008). Area distribution and scaling function for punctured polygons. Electronic Journal of Combinatorics, 15(1), Research Paper 53,50.

MLA:

Richard, Christoph, Iwan Jensen, and Anthony Guttmann. "Area distribution and scaling function for punctured polygons." Electronic Journal of Combinatorics 15.1 (2008): Research Paper 53,50.

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