Permutations in competing growth processes and balls-in-bins
Johannes Bäumler, Tejas Iyer
TL;DR
The authors study competing growth processes with independent, increasing value trajectories and random inter-jump times. They prove that in the non-leadership regime, every possible ranking of the agents occurs infinitely often, and in the exponential-waiting-time case this resolves a conjecture by Spencer for balls-in-bins with feedback via Rubin's construction. The core methods combine a zero-initial comparison, dispersion bounds for sums of independent increments, and probabilistic coupling to extend results to general initial conditions. The findings have implications for urn models, preferential attachment, and related reinforced growth processes, highlighting a universal property of non-leader dynamics. The results provide a rigorous link between ranking fluctuations and permutation-rich asymptotics in competing growth systems.
Abstract
Consider a model of $N$ independent, increasing $\mathbb{N}_0$-valued processes, with random, independent waiting times between jumps. It is known that there is either an emergent `leader', in which a single process possesses the maximal value for all sufficiently large times, or every pair of processes alternates leadership infinitely often. We show that in the latter regime, almost surely, one sees every possible permutation of rankings of processes infinitely often. In the case that the waiting times are exponentially distributed, this proves a conjecture from Spencer (appearing in a paper from Oliveira) on the `balls-in-bins' process with feedback [Conjecture 1, Combin. Probab. Comput. 17(1):87-110, (2008)].