Stabilization Time for a Type of Evolution on Binary Strings

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 $11\cdots 1100\cdots 00$, 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 $\{0,1,\ldots n-1\}$. We study the asymptotic behavior of this random variable as $n\rightarrow \infty$, and we determine its limit distribution in the weak sense after suitable centering and scaling. When $p \ne \frac{1}{2}$, the limit distribution is Gaussian. When $p = \frac{1}{2}$, 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 $p_n = \frac{1}{2}+ \frac{\lambda / 2}{\sqrt{n}}$ 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 $[0,1]$.