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.
Acknowledgements
M.C.’s research was supported by CONICET [grant number 10520170300561CO]. The authors are grateful to Andrew Wade for suggestions that improved the presentation.
Citation
Marcelo Costa. Jonathan Jordan. "Phase transitions in non-linear urns with interacting types." Bernoulli 28 (4) 2546 - 2562, November 2022. https://doi.org/10.3150/21-BEJ1428