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{\bf Christoph Richard, Iwan Jensen and Anthony J. Guttmann}
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{\bf Area Distribution and Scaling Function for Punctured Polygons}
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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.

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