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.