  We use the periodic Schur process, introduced in arXiv:math/0601019v1, to
study the random height function of lozenge tilings (equivalently, dimers) on
an infinite cylinder distributed under two variants of the
$q^{\operatorname{vol}}$ measure. Under the first variant, corresponding to
random cylindric partitions, the height function converges to a deterministic
limit shape and fluctuations around it are given by the Gaussian free field in
the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the
second variant, corresponding to an unrestricted dimer model on the cylinder,
the fluctuations are given by the same Gaussian free field with an additional
discrete Gaussian shift component. Fluctuations of the latter type have been
previously conjectured for dimer models on planar domains with holes.
