Every first year probability student knows that one may shed light on a joint probability law by computing its marginals, but that this does not determine the joint law uniquely. Conversely, one may gain insight about a given probability law by understanding the joint laws that have it as a marginal. Such arguments are generally referred to as coupling methods, and they are of vital importance in the study of various statistical mechanics models such as random walks and the Ising model. As such, it is of inherent interest to simply understand criteria under which couplings exist.

In this talk, we will discuss classical couplings between the Ising model and its graphical representations as well as try to give a flavour of the arguments that they help one accomplish. Finally, we will discuss a theorem which has all the classical couplings of the Ising model as special cases, and which yields at least three novel couplings.

Based on joint work with Jianping Jiang and Frederik Ravn Klausen.