Stabilization Time for a Type of Evolution on Binary Strings

TL;DR

This paper analyzes the stabilization time of a binary string evolution process, revealing its asymptotic distribution as the string length grows, with Gaussian and chi distributions depending on initial probabilities.

Contribution

It introduces a detailed asymptotic analysis of the stabilization time, including explicit limit distributions for different initial bit probabilities and a threshold setting.

Findings

Limit distribution is Gaussian for p ≠ 1/2.

Limit distribution is chi_3 for p = 1/2.

A family of distributions interpolates between chi_3 and Gaussian.

Abstract

We consider a type of evolution on {0,1}^n which occurs in discrete steps whereby at each step, we replace every occurrence of the substring "01" by "10". After at most n-1 steps we will reach a string of the form 11..1100..11, which we will call a "stabilized" string and we call the number of steps required the "stabilization time". If we choose each bit of the string independently to be a 1 with probability p and a 0 with probability 1-p, then the stabilization time of a string in {0,1}^n is a random variable with values in 0,1,...,n-1}. We study the asymptotic behavior of this random variable as n goes to infinity and we determine its limit distribution after suitable centering and scaling . When p is not 1/2, the limit distribution is Gaussian. When p = 1/2, the limit distribution is a \chi_3 distribution. We also explicitly compute the limit distribution in a threshold setting where p=p_n varies with n given by p_n = 1/2 + \lambda / 2 \sqrt{n} for \lambda > 0 a fixed parameter. This analysis gives rise to a one parameter family of distributions that fit between a \chi_3 and a Gaussian distribution. The tools used in our arguments are a natural interpretation of strings in {0,1}^n as Young diagrams, and a connection with the known distribution for the maximal height of a Brownian path on [0,1].