$\delta$-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 \textit{et al} 2015 {\it Phys. Rev.} E \textbf{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.