Stabilization Time for a Type of Evolution on Binary Strings

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⋯ 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,… n-1} . We study the asymptotic behavior of this random variable as n→∞ , and we determine its limit distribution in the weak sense after suitable centering and scaling. When p 1/2 , the limit distribution is Gaussian. When p = 1/2 , the limit distribution is a χ _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+ λ / 2/√(n) for λ > 0 a fixed parameter. This analysis gives rise to a one parameter family of distributions that fit between a χ _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] .