Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings

We prove and generalize a conjecture in arXiv:1610.0474(4) about the asymptotics of $\frac{1}{\sqrt{n!}}{f}^{\lambda \mathrm{/}\mu }$, where $f^{\lambda \mathrm{/}\mu }$ is the number of standard Young tableaux of skew shape $\lambda \mathrm{/}\mu$ which have stable limit shape under the $1\mathrm{/}\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.