The Annals of Probability

Rigorous results for the N K model

Richard Durrett and Vlada Limic

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Motivated by the problem of the evolution of DNA sequences, Kauffman and Levin introduced a model in which fitnesses were assigned to strings of 0's and 1's of length N based on the values observed in a sliding window of length $K+1$. When $K\ge 1$, the landscape is quite complicated with many local maxima. Its properties have been extensively investigated by simulation but until our work and the independent investigations of Evans and Steinsaltz little was known rigorously about its properties except in the case $K=N-1$. Here, we prove results about the number of local maxima, their heights and the height of the global maximum. Our main tool is the theory of (substochastic) Harris chains.

Article information

Ann. Probab., Volume 31, Number 4 (2003), 1713-1753.

First available in Project Euclid: 12 November 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60F05: Central limit and other weak theorems

NK model fitness local maxima limit theorems R-recurrence


Durrett, Richard; Limic, Vlada. Rigorous results for the N K model. Ann. Probab. 31 (2003), no. 4, 1713--1753. doi:10.1214/aop/1068646364.

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