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The phase transition for domination in two interacting urns with power reinforcement
Abstract
We study two interacting urns with power reinforcement W(n)=nα, for α>1. Using Qin's stochastic-approximation theorem as input, we reduce the phase transition problem for domination to the sign analysis of a single analytic function Φp. This deterministic reduction identifies the sharp threshold and proves that the two natural critical parameters for domination coincide. We determine the first-order behavior of the critical interaction parameter as α↓1 and show that it is not nondecreasing in the reinforcement exponent, answering two questions posed by Qin.
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BibTeX
@misc{xia2026phase,
title = {{The phase transition for domination in two interacting urns with power reinforcement}},
author = {Jiaming Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.22438},
eprint = {2607.22438},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.22438},
}
RIS
TY - UNPB
TI - The phase transition for domination in two interacting urns with power reinforcement
AU - Jiaming Xia
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2607.22438
UR - https://arxiv.org/abs/2607.22438
AB - We study two interacting urns with power reinforcement W(n)=n^α, for α>1. Using Qin's stochastic-approximation theorem as input, we reduce the phase transition problem for domination to the sign analysis of a single analytic function Φ_p. This deterministic reduction identifies the sharp threshold and proves that the two natural critical parameters for domination coincide. We determine the first-order behavior of the critical interaction parameter as α↓1 and show that it is not nondecreasing in the reinforcement exponent, answering two questions posed by Qin.
ER -