Abstract: The dimer model, i.e. random perfect matchings of a bipartite graph, is a classical object about which much is known. As soon as one biases the probability measure by edge weights which are themselves random, very little is known rigorously, though physicists have studied such models for several decades and made extensive predictions. I will discuss a new integrable model in this class (the Gamma-disordered Aztec diamond) which allows us to prove results on the free energy, and also exhibits surprising relations to integrable polymer models which lead to probabilistic results on tilings. Joint work with Maurice Duits (KTH), https://arxiv.org/abs/2512.03033.