Prudent walks are special self-avoiding walks that never take a step towards
an already occupied site, and \emph{k-sided prudent walks} (with k=1,2,3,4)
are, in essence, only allowed to grow along k directions. Prudent polygons
are prudent walks that return to a point adjacent to their starting point.
Prudent walks and polygons have been previously enumerated by length and
perimeter (Bousquet-M\'elou, Schwerdtfeger; 2010). We consider the enumeration
of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find
that the generating function is expressed in terms of a q-hypergeometric
function, with an accumulation of poles towards the dominant singularity. This
expression reveals an unusual asymptotic structure of the number of polygons of
area n, where the critical exponent is the transcendental number log23
and and the amplitude involves tiny oscillations. Based on numerical data, we
also expect similar phenomena to occur for 4-sided polygons. The asymptotic
methodology involves an original combination of Mellin transform techniques and
singularity analysis, which is of potential interest in a number of other
asymptotic enumeration problems.Comment: 27 pages, 6 figure