%0 Book Section %1 statphys23_0693 %A Jensen, I. %A Guttmann, A.J. %A Rechnitzer, A. %A Zabrocki, M. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K combinatorics differential enumerative equations exact fuchsian polygons self-avoiding solutions statphys23 topic-1 %T Exact solutions for punctured staircase polygons %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=693 %X In many cases real life phenomena are modelled by simplified solvable models, which despite the simplifications can give us great insight into the behaviour of the more complicated fully-fledged problem. A well-known long standing problem in statistical mechanics is to find the perimeter generating function for self-avoiding polygons on two-dimensional lattices. Several simplifications of this problem are solvable, but all the simpler models impose an effective directedness or other constraint that reduces the problem, in essence, to a one-dimensional problem. A very important and interesting insight gained from these simple models (staircase polygons in particular) is a conjecture for the limit distribution of area moments and scaling function for self-avoiding polygons. Here we report on the discovery of the exact perimeter generating function for two models of punctured staircase polygons. We started by counting the exact number of punctured polygons. Using this series we found that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation. In one case we managed to solve the ODE and find a closed form expression for the generating function. We have since been able to prove this results exactly using combinatorial arguments. This solution allows a generalisation to a model with any fixed number of nested punctures.

@incollection{statphys23_0693, abstract = {In many cases real life phenomena are modelled by simplified solvable models, which despite the simplifications can give us great insight into the behaviour of the more complicated fully-fledged problem. A well-known long standing problem in statistical mechanics is to find the perimeter generating function for self-avoiding polygons on two-dimensional lattices. Several simplifications of this problem are solvable, but all the simpler models impose an effective directedness or other constraint that reduces the problem, in essence, to a one-dimensional problem. A very important and interesting insight gained from these simple models (staircase polygons in particular) is a conjecture for the limit distribution of area moments and scaling function for self-avoiding polygons. Here we report on the discovery of the exact perimeter generating function for two models of punctured staircase polygons. We started by counting the exact number of punctured polygons. Using this series we found that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation. In one case we managed to solve the ODE and find a closed form expression for the generating function. We have since been able to prove this results exactly using combinatorial arguments. This solution allows a generalisation to a model with any fixed number of nested punctures.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Jensen, I. and Guttmann, A.J. and Rechnitzer, A. and Zabrocki, M.}, biburl = {https://www.bibsonomy.org/bibtex/2166d9c9d5416e086f5c7d21c40e74ad0/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {08fcb4f5d038c0b455877fe7a7c852f2}, intrahash = {166d9c9d5416e086f5c7d21c40e74ad0}, keywords = {combinatorics differential enumerative equations exact fuchsian polygons self-avoiding solutions statphys23 topic-1}, month = {9-13 July}, timestamp = {2007-06-20T10:16:27.000+0200}, title = {Exact solutions for punctured staircase polygons}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=693}, year = 2007 }