Some large polyominoe's perimeter: a stochastic analysis

TL;DR

This paper investigates the stochastic properties of various large polyominoes, demonstrating their perimeter distributions tend to Gaussian and Brownian motion as size increases, with all configurations considered equally likely.

Contribution

It provides the first detailed stochastic analysis of large polyomino perimeters, including Gaussian limits and Brownian motion convergence, for several polyomino classes.

Findings

Perimeter distributions are asymptotically Gaussian.

Perimeter trajectories converge to Brownian motion.

Markov properties of column chains are established.

Abstract

In this paper, we analyze the stochastic properties of some large size (area) polyominoe's perimeter such that the directed column-convex polyomino, the column-convex polyomino, the directed diagonally-convex polyomino, the staircase (or parallelogram) polyomino, the escalier polyomino, the wall (or bargraph) polyomino. All polyominoes considered here are made of contiguous, not-empty columns, without holes, such that each column must be adjacent to some cell of the previous column. We compute the asymptotic (for large size $n$) Gaussian distribution of the perimeter, including the corresponding Markov property of the chain of columns, and the convergence to classical Brownian motions of the perimeter seen as a trajectory according to the successive columns. All polyominoes of size $n$ are considered as equiprobable.