School Seminars and Colloquia

Area distribution and scaling function for punctured polygons

Statistical Mechanics/Combinatorics Seminar

by Dr Iwan Jensen

Institution: The University of Melbourne
Date: Thu 19th October 2006
Time: 3:15 PM
Location: Sisalkraft Theatre, Architecture (Ground floor),The University of Melbourne

Abstract: We study polygons with a finite number of punctures. 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. We rigorously analyse the effect of punctures on the area-distribution and obtain
expressions for the leading amplitudes of the area-moments for punctured polygons in
terms of the amplitudes for unpunctured polygons. For staircase polygons this leads
to exact formulas for the amplitudes. For self-avoiding polygons the formulas contain
certain constants which aren't known exactly but can be estimated numerically to a
very high degree of accuracy. Our analysis also leads to conjectures about the possible
scaling functions for the models. The expressions for the amplitudes are thoroughly checked numerically. For staircase polygons with up to 5 minimal punctures and staircase polygons with one or two punctures of fixed size we use series expansions to find exact generating functions for area-moments to order 10. In all cases the leading amplitudes agree with the proved formulas. Interestingly we find that the amplitude of the correction term is proportional to the corresponding leading amplitude with one less puncture. For staircase polygons with one and two punctures of arbitrary size
a careful asymptotic analysis of the series for the area-moments yields very accurate estimates for the amplitudes, again confirming the exact formulas. Finally, we also analyse series for
self-avoiding polygons with one and two punctures (minimal as well as arbitrary). In this case the numerical evidence is not quite as convincing, but we do find that the numerical estimates agree with the exact formulas to at least 3-4 significant digits.

This is work with C. Richard (Universitaet Bielefeld) and our very own A. J. Guttmann

For More Information: Dr. Iwan Jensen Phone: +61 3 8344 5214