$\delta$-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 $Y_k>Y_{k-1}-{\delta }_{k-1}$ with a deterministic sequence ${\delta }_k>0$ called the handicap. For constant $\delta_k \equiv \delta$ and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of $\delta$ 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 $\mathit{\text{et}}\mathit{\text{al}}$ 2015 {\it Phys. Rev.} E $\mathbf{\text{91}}$ 042707). Here we explore the behavior for general probability distributions and decreasing and increasing sequences ${\delta }_k$, focusing in particular on the case when ${\delta }_k$ 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 ${\delta }_k$ is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where ${\delta }_k$ 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.