We study the inflated phase of two dimensional lattice polygons, both convex and column-convex, with fixed area A and variable perimeter, when a weight μ exp[−Jb] is associated to a polygon with perimeter t and b bends. The mean perimeter is calculated as a function of the fugacity μ and the bending rigidity J . In the limit μ → 0, the mean perimeter has the asymptotic behaviour 〈t〉/4 √ A ≃ 1−K...

An explicit expression is obtained for the perimeter and area generating function G(y, z) = ∑ n>=2 ∑ m>=1 cn,my z, where cn,m is the number of row-convex polygons with area m and perimeter n. A similar expression is obtained for the area-perimeter generating function for staircase polygons. Both expressions contain q-series.

We study the conformal capacity by using novel computational algorithms based on implementations of fast multipole method, and analytic techniques. Especially, we apply domain functionals to capacities condensers $(G,E)$ where $G$ is a simply connected in complex plane $E$ compact subset $G$. Due invariance, our main tools are hyperbolic geometry such as perimeter $E$. Our experiments demonstra...