Abstract
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.
Original language | English (US) |
---|---|
Pages (from-to) | 328-340 |
Number of pages | 13 |
Journal | Stochastic Models |
Volume | 29 |
Issue number | 3 |
DOIs | |
State | Published - Jul 3 2013 |
Keywords
- Discrete logistic equation
- Evolution
- Fixation time
- Galton-Watson scaling limit
ASJC Scopus subject areas
- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics