Phase transitions in non-linear urns with interacting types

DOI10.3150/21-BEJ1428arXiv2006.02685WikidataQ114038740 ScholiaQ114038740MaRDI QIDQ6342084

Marcelo Azevedo Costa, Jonathan H. Jordan

Publication date: 4 June 2020

Abstract: We investigate reinforced non-linear urns with interacting types, and show that where there are three interacting types there are phenomena which do not occur with two types. In a model with three types where the interactions between the types are symmetric, we show the existence of a double phase transition with three phases: as well as a phase with an almost sure limit where each of the three colours is equally represented and a phase with almost sure convergence to an asymmetric limit, which both occur with two types, there is also an intermediate phase where both symmetric and asymmetric limits are possible. In a model with anti-symmetric interactions between the types, we show the existence of a phase where the proportions of the three colours cycle and do not converge to a limit, alongside a phase where the proportions of the three colours can converge to a limit where each of the three is equally represented.

Mathematics Subject Classification ID

Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) (60J20)

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