  Ultracold polar molecules, in highly anisotropic traps and interacting via a
repulsive dipolar potential, may form one-dimensional chains at high densities.
According to classical theory, at low temperatures there exists a critical
value of the density at which a second order phase transition from a linear to
a zigzag chain occurs. We study the effect of thermal and quantum fluctuations
on these self-organized structures using classical and quantum Monte Carlo
methods, by means of which we evaluate the pair correlation function and the
static structure factor. Depending on the parameters, these functions exhibit
properties typical of a crystalline or of a liquid system. We compare the
thermal and the quantum results, identifying analogies and differences.
Finally, we discuss experimental parameter regimes where the effects of quantum
fluctuations on the linear - zigzag transition can be observed.
