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.

The logarithm of a unitary characteristic polynomial is a Gaussian random variable when one draws the matrix under Haar measure. Large deviation principles and some precise deviations for this random variable are known following work of Hughes-Keating-O’Connell and Féray-Méliot-Nikeghbali. Motivated by a conjecture regarding the leading coefficient of the moments of the Riemann zeta function, we study a particular ‘interpolating regime’ in the right-tail. This is joint work with Sebastian Ortiz.