The perimeter of uniform and geometric words: a probabilistic analysis

TL;DR

This paper analyzes the perimeter of random words modeled as polyominoes, deriving moments, asymptotic Gaussian distribution, and convergence to Brownian motion using probabilistic methods.

Contribution

It introduces a probabilistic framework for computing perimeter moments of uniform and geometric words and establishes their asymptotic Gaussian behavior and convergence to Brownian motion.

Findings

Perimeter moments are explicitly computed for uniform and geometric words.

Perimeter distribution converges to a Gaussian distribution asymptotically.

Perimeter process converges in distribution to a Brownian motion.

Abstract

Let a word be a sequence of $n$ i.i.d. integer random variables. The perimeter $P$ of the word is the number of edges of the word, seen as a polyomino. In this paper, we present a probabilistic approach to the computation of the moments of $P$. This is applied to uniform and geometric random variables. We also show that, asymptotically, the distribution of $P$ is Gaussian and, seen as a stochastic process, the perimeter converges in distribution to a Brownian motion