Stochastic Flips on Two-letter Words

TL;DR

This paper studies a Markov process on two-letter words involving local flips that do not increase consecutive identical pairs, analyzing the expected convergence time to perfectly alternating words.

Contribution

It introduces a new flip-based Markov process model for two-letter words and provides bounds on the expected convergence time in worst-case and average-case scenarios.

Findings

Expected convergence time is O(n^3) in the worst case.

Expected convergence time is O(n^{5/2} log n) on average.

The process models quasicrystal growth phenomena.

Abstract

This paper introduces a simple Markov process inspired by the problem of quasicrystal growth. It acts over two-letter words by randomly performing \emph{flips}, a local transformation which exchanges two consecutive different letters. More precisely, only the flips which do not increase the number of pairs of consecutive identical letters are allowed. Fixed-points of such a process thus perfectly alternate different letters. We show that the expected number of flips to converge towards a fixed-point is bounded by $O(n^3)$ in the worst-case and by $O(n^{5/2}\ln{n})$ in the average-case, where $n$ denotes the length of the initial word.