δ-exceedance records and random adaptive walks

Abstract

We study a modified record process where the kth record in a series of independent and identically distributed random variables is defined recursively through the condition with a deterministic sequence called the handicap. For constant 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 , focusing in particular on the case when 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 is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where 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.