  We present an exact solution for the distribution of sample averaged monomer
to monomer distance of ring polymers. For non-interacting and
weakly-interacting models these distributions correspond to the distribution of
the area under the reflected Bessel bridge and the Bessel excursion
respectively, and are shown to be identical in dimension d greater or equal 2.
A symmetry of the problem reveals that dimension d and 4 minus d are
equivalent, thus the celebrated Airy distribution describing the areal
distribution of the one dimensional Brownian excursion describes also a polymer
in three dimensions. For a self-avoiding polymer in dimension d we find
numerically that the fluctuations of the scaled averaged distance are nearly
identical in dimensions 2 and 3, and are well described to a first
approximation by the non-interacting excursion model in dimension 5.
