$\delta$-exceedance records and random adaptive walks

We study a modified record process where the k’th record in a series of independent and identically distributed random variables is defined recursively through the condition Yk > Yk−1 − δk−1 with a deterministic sequence δk > 0 called the handicap. For constant δk ≡ δ and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of δ between a normal phase where the mean record value increases indefinitely and a stationary phase where the mean record value remains bounded and a finite fraction of all entries are records (Park et al 2015 Phys. Rev. E 91 042707). Here we explore the behavior for general probability distributions and decreasing and increasing sequences δk, focusing in particular on the case when δk matches the typical spacing between subsequent records in the underlying simple record process without handicap. We find that a continuous phase transition occurs only in the exponential case, but a novel kind of first order transition emerges when δk is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where δk corresponds to the change of the deterministic fitness component after k mutational steps. The results for the record process are used to compute the mean number of steps that a population performs in such a landscape before being trapped at a local fitness maximum.