Phase transitions from exp⁡(n^1/2) to exp⁡(n^2/3) 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 ₁n ⁻¹exp⁡(c ₂n ^1/2) to c ₃n ^−49/72exp⁡(c ₄n ^2/3+c ₅n ^1/3) for some explicit coefficients c ₁,…,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.