When does a highly-polarised debate lead to a breakdown in communication? How can conflicts be resolved in these situations? Is it more effective for political parties spend their funds on broadly targeted messages, or persuasive local campaigning? In this talk I will describe a “statistical physics” approach to these questions, formulating simple and highly-stylised models which can be analysed mathematically. I will show how the cumulative effect of random fluctuations in these models plays a crucial role in determining the outcome.