Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2006-10-23
J. Phys. A: Math. Theor. 41 (2008) 215002
Physics
Condensed Matter
Statistical Mechanics
14 pages, 2 figures, IoP style files. Extended version with Andrew Rechnitzer including combinatorial proof and many new resul
Scientific paper
We have derived the perimeter generating function of a model of punctured staircase polygons in which the internal staircase polygon is rotated by a 90degree angle with respect to the outer staircase polygon. In one approach we calculated a long series expansion for the problem and found that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 4. We then solved this ODE and found a closed form expression for the generating function. This is a highly unusual and most fortuitous result since ODEs of such high order very rarely permit a closed form solution. In a second approach we proved the result for the generating function exactly using combinatorial arguments. This latter solution allows many generalisations including to models with other types of punctures and to a model with any fixed number of nested rotated staircase punctures.
Jensen Iwan
Rechnitzer Andrew
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