Suppose, I begin with a single string $X=\{ X_1, X_2, .... X_n\}$ where all $X_i$ are initialized to zero. Now, I evolve the system under the following rules:
1) At each time step, a string can be exactly copied into another string. A string can only be copied once each step and any string can be copied only $k$ times. If $k \rightarrow \infty$, then $N(t)=2^t$. Here, $N(t)$ denotes no of strings at time $t$. If a string has not been copied k times, its elements evolve via the following rules in time:
2) $P(X_i(t+1)=1|X_i(t)=0)=q$. Here, t means any discrete time step and q is the probability of flip from 0 to 1.
3) $P(X_i(t+1)=0|X_i(t)=0)=1-q$.
4) $P(X_i(t+1)=1|X_i(t)=1)=1$.
5) Once a string has been copied $k$ times, its elements freeze in time.
The question is: What is the average hamming distance(HD) between 2 strings after time $t$ of system evolution?
PS: I can exactly solve the problem for $k \rightarrow \infty$ and the answer is $\langle HD(t) \rangle=2x_t(1-x_t)$ where $x_t = (1-q)^t$ and $\langle...\rangle$ denote the average. I need to solve it for finite $k$.
Any help in this regard would be of much help. Thanks.