Phase transitions from $\exp (n^{1\mathrm{/}2})$ to $\exp (n^{2\mathrm{/}3})$ in the asymptotics of banded plane partitions

We examine the asymptotics of a class of banded plane partitions under a varying bandwidth parameter $m$, and clarify the transitional behavior for large size $n$ and increasing $m=m(n)$ to be from $c_1 n^{-1} \exp(c_2 n^{1/2})$to$c_3 n^{-49/72} \exp(c_4 n^{2/3} + c_5 n^{1/3})$ for some explicit coefficients $c_1,\dots ,c_5$. The method of proof, which is a unified saddle-point analysis for all phases, is general and can be extended to other classes of plane partitions.