The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics

TL;DR

This paper studies the enumeration of prudent polygons by area, revealing unusual asymptotic behavior with transcendental critical exponents and oscillations, using advanced Mellin transform and singularity analysis techniques.

Contribution

It provides the first explicit enumeration and asymptotic analysis of prudent polygons by area, uncovering novel asymptotic phenomena and methodological approaches.

Findings

Generating function expressed via $q$-hypergeometric function.

Critical exponent is the transcendental number $ ext{log}_2 3$.

Oscillatory behavior in the asymptotics of polygon counts.

Abstract

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 $\log_23$ 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.