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The $\varphi^4$ model is a generalisation of the Ising model to a system with unbounded spins that are confined by a quartic potential. Its significance in statistical physics was first noted by Griffiths and Simon, who observed that the $\varphi^4$ potential arises as the scaling limit of the fluctuations of the critical Ising model on the complete graph. In this talk, I will describe how this connection to the Ising model leads to two new geometric representations of the $\varphi^4$ model, called the random tangled current expansion and the random cluster model. I will explain how these representations can be used to prove that the phase transition of the $\varphi^4$ model is continuous in dimensions three and higher, and to obtain large-deviation estimates for spin averages in the supercritical regime. Based on joint works with Trishen Gunaratnam, Romain Panis, and Franco Severo.
