Self-similarity in an exchangeable site-dynamics model

Abstract

We consider a model for which every site of $\mathbb{N}$ is assigned a fitness in $[0,1]$. At every discrete time all the sites are updated and each site samples a uniform on $[0,1]$, independently of everything else. At every discrete time and independently of the past the environment is good with probability $p$ or bad with probability $1-p$. The fitness of each site is then updated to the maximum or the minimum between its present fitness and the sampled uniform, according to whether the environment is good or bad. Assuming the initial fitness distribution is exchangeable over the site indexing, the empirical fitness distribution is a probability-valued Markov process. We show that this Markov process converges to an explicitly-identified stationary distribution exhibiting a self-similar structure.