We study a class of close-packed dimer models on the square lattice, in the presence of small but extensive perturbations that make them non-determinantal. Examples include the 6-vertex model close to the free-fermion point, and the dimer model with
plaquette interaction previously analyzed in cite{A,AL,GMT17a,GMT17b}. By tuning the edge weights, we can impose a non-zero average tilt for the height function, so that the considered models are in general not symmetric under discrete rotations and reflections. In the determinantal case, height fluctuations in the massless (or `liquid) phase scale to a Gaussian log-correlated field and their amplitude is a universal constant, independent of the tilt. When the perturbation strength $lambda$ is sufficiently small we prove, by fermionic constructive Renormalization Group methods, that log-correlations survive, with amplitude $A$ that, generically, depends non-trivially and non-universally on $lambda$ and on the tilt. On the other hand, $A$ satisfies a universal scaling relation (`Haldane or `Kadanoff relation), saying that it equals the anomalous exponent of the dimer-dimer correlation.
We consider a system of anisotropic plates in the three-dimensional continuum, interacting via purely hard core interactions. We assume that the particles have a finite number of allowed orientations. In a suitable range of densities, we prove the ex
istence of a uni-axial nematic phase, characterized by long range orientational order (the minor axes are aligned parallel to each other, while the major axes are not) and no translational order. The proof is based on a coarse graining procedure, which allows us to map the plate model into a contour model, and in a rigorous control of the resulting contour theory, via Pirogov-Sinai methods.
We consider a three-dimensional chaotic system consisting of the suspension of Arnolds cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relat
ive phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.
