Phase Transition in Conditional Curie-Weiss Model

Abstract: This paper proposes a conditional Curie-Weiss model as a model for opinion formation in a society polarized along two opinions, say opinions 1 and 2. The model comes with interaction strength $\beta>0$ and bais $h$. Here the population in question is divided into three main groups, namely: Group one consisting of individuals who have decided on opinion 1. Let the proportion of this group be given by $s$. Group two consisting of individauls who have chosen opinion 2. Let $r$ be their proportion. Group three consisting of individuals who are yet to decide and they will decide based on their environmental conditions. Let $1-s-r$ be the proportion of this group. We show that the specific magnetization of the associated conditional Curie-Weiss model has a first order phase transition (discontinuous jump in specific magnetization) at $\beta^*=\left(1-s-r\right)^{-1}$. It is also shown that not all the discontinuous jumps in magnetization will result in phase change. We point out how an extention of this model could serve as a random field Curie-Weiss model where the random field distribution has nonvanishing mean.