Exact and limit distributions of the largest fitness on correlated fitness landscapes

TL;DR

This paper analyzes the distribution of maximum fitness in correlated fitness landscapes, providing exact and limit distributions for various correlation regimes, revealing deviations from the standard Gumbel distribution.

Contribution

It offers analytical solutions for the maximum fitness distribution in models with different correlation structures, extending understanding beyond independent cases.

Findings

Limit distributions deviate from Gumbel form

Analytical results for three correlation regimes

Correlations significantly affect fitness maxima

Abstract

We study the distribution of the maximum of a set of random fitnesses with fixed number of mutations in a model of biological evolution. The fitness variables are not independent and the correlations can be varied via a parameter $\ell=1,...,L$. We present analytical calculations for the following three solvable cases: (i) one-step mutants with arbitrary $\ell$ (ii) weakly correlated fitnesses with $\ell=L/2$ (iii) strongly correlated fitnesses with $\ell=2$. In all these cases, we find that the limit distribution for the maximum fitness is not of the standard Gumbel form.