## The Annals of Probability

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

### Rigorous results for the *N K* model

Richard Durrett and Vlada Limic

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 November 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1068646364

**Digital Object Identifier**

doi:10.1214/aop/1068646364

**Mathematical Reviews number (MathSciNet)**

MR2016598

**Zentralblatt MATH identifier**

1049.60037

**Subjects**

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

**Keywords**

NK model fitness local maxima limit theorems R-recurrence

#### Citation

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