Competing growth processes with random growth rates and random birth times

Comparing individual contributions in a strongly interacting system of stochastic growth processes can be a very difficult problem. This is particularly the case when new growth processes are initiated depending on the state of previous ones and the growth rates of the individual processes are themselves random. We propose a novel technique to deal with such problems and show how it can be applied to a broad range of examples where it produces new insight and surprising results. The method relies on two steps: In the first step, which is highly problem dependent, the growth processes are jointly embedded into continuous time so that their evolutions after initiation become approximately independent while we retain some control over the initiation times. Once such an embedding is achieved, the second step is to apply a Poisson limit theorem that enables a comparison of the state of the processes initiated in a critical window and therefore allows an asymptotic description of the extremal process. In this paper we prove a versatile limit theorem of this type and show how this tool can be applied to obtain novel asymptotic results for a variety of interesting stochastic processes. These include (a) the maximal degree in different types of preferential attachment networks with fitnesses like the well-known Bianconi–Barabási tree and a network model of Dereich, (b) the most successful mutant in branching processes evolving by selection and mutation, and (c) the ratio between the largest and second largest cycles in a random permutation with random cycle weights, which can also be interpreted as a disordered version of Pitman's Chinese restaurant process.