Formation of consensus, in binary yes/no type of voting, is a well defined process. However, even in presence of clear incentives, the dynamics involved can be incredibly complex. Specifically, formations of large groups of similarly opinionated individuals could create a condition of `support-bubbles’ or spontaneous polarization that renders consensus virtually unattainable (e.g., the question of the UK exiting the EU). There have been earlier attempts in capturing the dynamics of consensus formation in societies through simple $Z_2$-symmetric models hoping to capture the essential dynamics of average behavior of a large number of individuals in a statistical sense. However, in absence of external noise, they tend to reach a frozen state with fragmented and polarized states i.e., two or more groups of similarly opinionated groups with frozen dynamics. Here we show in a kinetic exchange opinion model (KEM) considered on $L\times L$ square lattices, that while such frozen states could be avoided, an exponentially slow approach to consensus is manifested. Specifically, the system could either reach consensus in a time that scales as $L^2$ or a long lived metastable state (termed a domain-wall state) for which formation of consensus takes a time scaling as $L^{3.6}$. The latter behavior is comparable to some voter-like models with intermediate states studied previously. The late-time anomaly in the time scale is reflected in the persistence probability of the model. Finally, the interval of zero-crossing of the average opinion i.e., the time interval over which the average opinion does not change sign is shown to follow a scale free distribution, which is compared with that seen in the opinion surveys regarding Brexit and associated issues in the last 40 years. The issue of minority spreading is also addressed by calculating the exit probability.

中文翻译：

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达成共识的漫长道路：二元选择投票模型的两阶段粗化

以二进制“是/否”投票方式形成共识是一个明确定义的过程。但是，即使有明确的激励措施，所涉及的动力也可能非常复杂。具体而言，由大量持类似观点的个人组成的团体可能会造成“支持泡沫”或自发两极分化的状况，这实际上使达成共识几乎无法达成（例如，英国退出欧盟的问题）。以前有过尝试通过简单的方法捕获社会共识形成的动力。${ž}_2$对称模型，希望从统计意义上捕捉大量个体平均行为的基本动态。但是，在没有外部噪声的情况下，它们趋于达到具有零碎和极化状态的冻结状态，即两组或更多组具有相似动态的相似观点的组。在这里，我们展示了一个考虑以下因素的动力学交换意见模型（KEM）：$\mathrm{大号}\times \mathrm{大号}$正方形晶格，虽然可以避免这种冻结状态，但是却显示出以指数级缓慢的方式达成共识。具体来说，该系统可以在以下时间段内达成共识：${\mathrm{大号}}^2$ 或长期稳定的亚稳状态（称为域壁状态），为此形成共识需要时间缩放 ${\mathrm{大号}}^{3.6}$。后者的行为与之前研究过的具有中间状态的某些类似投票者的模型相当。时间尺度上的晚期异常反映在模型的持久性概率中。最后，平均意见的零交叉间隔（即，平均意见不变的时间间隔）显示为遵循无标度分布，并将其与关于英国退欧及相关问题的民意调查中所见相比较。在过去的40年中。还可以通过计算退出概率来解决少数群体扩散的问题。