Accessibility percolation on n-trees

–Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in ascending order. For the case when the random variables are independent and identically distributed, we derive an asymptotically exact expression for the probability that there is at least one accessible path from the root to the leaves in an n-tree. This probability tends to 1 (0) if the branching number is increased with the height of the tree faster (slower) than linearly. When the random variables are biased such that the mean value increases linearly with the distance from the root of the tree, a percolation threshold emerges at a finite value of the bias. Introduction and outline. – Percolation theory in its modern form was introduced by Broadbent and Ham-mersley in 1957 [1]. Since then percolation has become a cornerstone of probability theory and statistical physics, with applications ranging from molecular dynamics to star formation [2, 3], and new variants of the problem continue to attract much attention (e.g., [4–7]). Standard perco-lation theory is concerned with the loss of global connec-tivity in a graph when vertices or bonds are randomly removed , as quantified by the probability for the existence of an infinite cluster of contiguous vertices. Here we consider a novel kind of percolation problem inspired by evolutionary biology. Imagine a population of some lifeform endowed with the same genetic type (genotype). If a mutation occurs, a new genotype is created which can die out or replace the old one. Provided natural selection is sufficiently strong, the latter only happens if the new genotype has larger fitness. As a consequence, on longer timescales the genotype of the population takes a path through the space of genotypes along which the fitness is monotonically increasing [8]. Such a path is called selectively accessible [9, 10]. Since the relationship between genotype and fitness is very complicated and largely unknown, it is natural to assign fitness values to genotypes in a random way. Evolutionary accessibility thus becomes a statistical property of random fitness landscapes [11–13]. In recent years it has become possible to experimentally determine fitness land