The perimeter of uniform and geometric words: a probabilistic analysis

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.

Mathematics Subject Classification (2010): 05A16, 05A05, 60C05, 60F05.

Keywords: Words, perimeter, moments, probabilistic approach, Gaussian distribution, Brownian motion