Phase transitions in non-linear urns with interacting types

TL;DR

This paper explores phase transitions in non-linear urn models with three interacting types, revealing complex behaviors such as multiple phases, symmetric and asymmetric limits, and cyclic proportions not seen with fewer types.

Contribution

It introduces new phenomena in three-type interacting urns, including double phase transitions and cyclic behaviors, expanding understanding beyond two-type models.

Findings

Existence of a double phase transition with three phases in symmetric interactions.

Identification of cyclic proportion behavior in anti-symmetric interactions.

Discovery of intermediate phase with both symmetric and asymmetric limits.

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.