  We consider a four-vertex model introduced by B\'{a}lint T\'{o}th: a
dependent bond percolation model on $\mathbb{Z}^2$ in which every edge is
present with probability 1/2 and each vertex has exactly two incident edges,
perpendicular to each other. We prove that all components are finite cycles
almost surely, but the expected diameter of the cycle containing the origin is
infinite. Moreover, we derive the following critical exponents: the tail
probability $\mathbb{P}$(diameter of the cycle of the origin $>$$n$) $\approx$
$n^{-\gamma}$ and the expectation $\mathbb{E}$(length of a typical cycle with
diameter $n)\approx n^{\delta}$, with $\gamma=(5-\sqrt{17})/4=0.219...$ and
$\delta=(\sqrt{17}+1)/4=1.28....$ The value of $\delta$ comes from a singular
sixth order ODE, while the relation $\gamma+\delta=3/2$ corresponds to the fact
that the scaling limit of the natural height function in the model is the
additive Brownian motion, whose level sets have Hausdorff dimension 3/2. We
also include many open problems, for example, on the conformal invariance of
certain linear entropy models.
