Phase transition in random adaptive walks on correlated fitness landscapes

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.