  Caustic singularities of the spatial distribution of particles in turbulent
aerosols enhance collision rates and accelerate coagulation. Here we
investigate how and where caustics form at weak particle inertia, by analysing
a three-dimensional Gaussian statistical model for turbulent aerosols in the
persistent limit, where the flow varies slowly compared with the particle
relaxation time. In this case, correlations between particle- and
fluid-velocity gradients are strong, and caustics are induced by large,
strain-dominated excursions of the fluid-velocity gradients. These excursions
must cross a characteristic threshold in the plane spanned by the invariants
$Q$ and $R$ of the fluid-velocity gradients. Our method predicts that the most
likely way to reach this threshold is by a unique ``optimal fluctuation'' that
propagates along the Vieillefosse line, $27R^2/4 +Q^3=0$. We determine the
shape of the optimal fluctuation as a function of time and show that it is
dominant in numerical statistical-model simulations even for moderate particle
inertia.
