Phase transition in random adaptive walks on correlated fitness landscapes

Abstract

We study biological evolution on a random fitness landscape where correlations are introduced through a linear fitness gradient of strength $c$. When selection is strong and mutations rare the dynamics is a directed uphill walk that terminates at a local fitness maximum. We analytically calculate the dependence of the walk length on the genome size $L$. When the distribution of the random fitness component has an exponential tail we find a phase transition of the walk length $D$ between a phase at small $c$ where walks are short $(D\sim \ln L)$ and a phase at large $c$ where walks are long $(D\sim L)$. For all other distributions only a single phase exists for any $c>0$. The considered process is equivalent to a zero temperature Metropolis dynamics for the random energy model in an external magnetic field, thus also providing insight into the aging dynamics of spin glasses.

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• Genotypic complexity of Fisher's geometric model
• Greedy adaptive walks on a correlated fitness landscape
• Accessibility percolation in random fitness landscapes
• Statistical properties of avalanches via the c-record process
• $\delta$-exceedance records and random adaptive walks