Fixation time for an evolutionary model

We study the asymptotic value as L → ∞ of the time for evolution τ, understood as the first time to reach a preferred word of length L using an alphabet with N letters. The word is updated at unit time intervals randomly, but configurations with letters matching with the preferred word are sticky, i.e., the probability to leave the configuration equals 0 ≤ γ ≤1, where γ may depend on the configuration. The model is introduced in Ref.^{[ 5 ]} in the case γ = 0, where it was shown that E[τ] ∼ Nln (L). We first give an alternative proof of the logarithmic scale, by evaluating the mode of τ. We then answer positively a question posed by H. Wilf on whether τ is exponential when γ ≠ 0. The natural scaling γ =O(L ^-1) gives rise to several finite order limits, including the interacting model when γ depends linearly on the number of matches with the preferred word. The scaling limit of the number of non-matching letters follows a Galton-Watson process with immigration. In a related model, the empirical measure converges to the solution of a discrete logistic equation with possible nonzero steady state. In conclusion, the length of τ is a question of scaling.