Exact perimeter generating function for a model of punctured staircase polygons
arXiv:cond-mat/0610605 · doi:10.1088/1751-8113/41/21/215002
Abstract
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.
14 pages, 2 figures, IoP style files. Extended version with Andrew Rechnitzer including combinatorial proof and many new results
References in corpus (8)
- The Fuchsian differential equation of the square lattice Ising model susceptibility
- Scaling behaviour of two-dimensional polygon models
- Scaling function and universal amplitude combinations for self-avoiding polygons
- Square lattice Ising model susceptibility: Series expansion method and differential equation for
- Ising model susceptibility: Fuchsian differential equation for and its factorization properties
- The perimeter generating function of punctured staircase polygons
- Fuchsian differential equation for the perimeter generating function of three-choice polygons
- Area distribution and scaling function for punctured polygons