Area Distribution and Scaling Function for Punctured Polygons

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. Based on an assumption about the limiting area distribution for unpunctured polygons, we rigorously analyse the effect of a finite number of punctures on the limiting area distribution in a uniform ensemble, where punctured polygons with equal perimeter have the same probability of occurrence. Our analysis leads to conjectures about the scaling behaviour of the models. We also analyse exact enumeration data. For staircase polygons with punctures of fixed size, this yields explicit expressions for the generating functions of the first few area moments. For staircase polygons with punctures of arbitrary size, a careful numerical analysis yields very accurate estimates for the area moments. Interestingly, we find that the leading correction term for each area moment is proportional to the corresponding area moment with one less puncture. We finally analyse corresponding quantities for punctured self-avoiding polygons and find agreement with the conjectured formulas to at least 3–4 significant digits.