  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].
