  We show there is a last path at the rough smooth boundary of the two-periodic
Aztec diamond with parameter $a\in (0,1)$ that, suitably rescaled, converges to
the Airy process, under the condition that $a$ tends to zero as the size of the
Aztec diamond tends to infinity at a certain rate. This condition causes the
rough region to have a thin, mesoscopic width. We also show that the dimers are
described by a discrete Bessel kernel when the width is only of microscopic
size.
