Asymptotic Analysis and Random Sampling of Digitally Convex Polyominoes

TL;DR

This paper provides a combinatorial analysis and a uniform sampling method for digitally convex polyominoes, revealing their limit shape and asymptotic properties through theoretical and experimental approaches.

Contribution

It introduces a new combinatorial description and a uniform sampling algorithm for digitally convex polyominoes, along with analysis of their asymptotic behavior and limit shape.

Findings

Sampler demonstrates a limit shape for large polyominoes

Derived combinatorial symbolic description of digitally convex polyominoes

Analyzed asymptotic properties of digitally convex polyominoes

Abstract

Recent work of Brlek \textit{et al.} gives a characterization of digitally convex polyominoes using combinatorics on words. From this work, we derive a combinatorial symbolic description of digitally convex polyominoes and use it to analyze their limit properties and build a uniform sampler. Experimentally, our sampler shows a limit shape for large digitally convex polyominoes.