论文标题
具有相互作用类型的非线性urn的相变
Phase transitions in non-linear urns with interacting types
论文作者
论文摘要
我们研究了具有相互作用类型的增强的非线性urn，并表明有三种相互作用的现象在两种类型中都不会发生。在具有三种类型的模型中，类型之间的相互作用是对称的，我们显示了具有三个阶段的双相转换的存在：以及几乎确保限制的相位，其中三种颜色中的每一种都同样代表，并且几乎可以确保与非对称限制的阶段相同，这两种类型都与两种类型发生，这两个类型也存在着中间相，并且可能存在对称性和随意的极限。在类型之间具有反对称相互作用的模型中，我们显示了一个相的存在，其中三种颜色循环的比例并且不会收敛到极限，而三种颜色的比例可以将三种颜色的比例收敛到限制，而三种颜色的每个颜色都同样表示。
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.
