$\delta$-exceedance records and random adaptive walks

Abstract

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.

minor changes. Published

References in corpus (2)

• Phase transition in random adaptive walks on correlated fitness landscapes
• Asymptotic normality for the counting process of weak records and δ-records in discrete models

Cited by in corpus (5)

• Extreme value statistics of correlated random variables: a pedagogical review
• Record statistics for random walks and Lévy flights with resetting
• Universality classes of interaction structures for NK fitness landscapes
• Genotypic complexity of Fisher's geometric model
• Statistical properties of avalanches via the c-record process