  We consider the dimer model on the Aztec diamond with Fock's weights, which
is gauge equivalent to the model with any choice of positive weight function.
We prove an explicit, compact formula for the inverse Kasteleyn matrix, thus
extending numerous results in the case of periodic graphs. We also show an
explicit product formula for the partition function; as a specific instance of
the genus 0 case, we recover Stanley's formula. We then use our explicit
formula for the inverse Kasteleyn matrix to recover, in a simple way, limit
shape results; we also obtain new ones. In doing so, we extend the
correspondence between the limit shape and the amoeba of the corresponding
spectral curve of arXiv:2306.07482 to the case of non-generic weights.
