It is shown that a recently conjectured form for the
critical scaling function for planar self-avoiding polygons
weighted by their perimeter and area
also follows from an exact renormalization group flow into the
branched polymer problem, combined with the dimensional reduction arguments of
Parisi and Sourlas. The result is generalized to
higher-order multicritical points, yielding exact values for all
their critical exponents and exact forms for the associated
scaling functions.
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