WIAS Preprint No. 3220, (2025)

Permutations in competing growth processes and balls-in-bins


Authors

  • Bäumler, Johannes
    ORCID: 0009-0005-6913-4992
  • Iyer, Tejas
    ORCID: 0000-0002-2005-3164

2020 Mathematics Subject Classification

  • 60G51 60J74 91B70

Keywords

  • Growth processes, birth processes, balls-in-bins processes with feedback, generalised Pólya urns, non-linear urns, convergence of random series, reinforced processes

DOI

10.20347/WIAS.PREPRINT.3220

Abstract

Consider a model of N independent, increasing ℕ0-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 [8, Conjecture1].

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