Several objects from discrete geometry may be seen as dynamical systems: an example is a discrete version of holomorphic functions via Shramm's orthogonal circle patterns, for which "analytic continuation" gives rise to discrete time dynamics. We will see that this evolution may as well be seen as a birational map, and that its iterates are encoded by the partition function of a model from statistical mechanics, the dimer model. This correspondence allows us to answer some questions about the initial dynamics, such as the study of its singularities. Joint works with Niklas Affolter and Béatrice de Tilière.
