### 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) |
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Pages (from-to) | 328-340 |

Number of pages | 13 |

Journal | Stochastic Models |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - Jul 3 2013 |

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### Keywords

- Discrete logistic equation
- Evolution
- Fixation time
- Galton-Watson scaling limit

### ASJC Scopus subject areas

- Modeling and Simulation
- Statistics and Probability
- Applied Mathematics

### Cite this

*Stochastic Models*,

*29*(3), 328-340. https://doi.org/10.1080/15326349.2013.808908