Integrable structure of melting crystal model with two q-parameters

This paper explores integrable structures of a generalized melting crystal model that has two $q$-parameters $q_1,q_2$. This model, like the ordinary one with a single $q$-parameter, is formulated as a model of random plane partitions (or, equivalently, random 3D Young diagrams). The Boltzmann weight contains an infinite number of external potentials that depend on the shape of the diagonal slice of plane partitions. The partition function is thereby a function of an infinite number of coupling constants $t_1,t_2,…$ and an extra one $Q$. There is a compact expression of this partition function in the language of a 2D complex free fermion system, from which one can see the presence of a quantum torus algebra behind this model. The partition function turns out to be a tau function (times a simple factor) of two integrable structures simultaneously. The first integrable structure is the bigraded Toda hierarchy, which determine the dependence on $t_1,t_2,…$. This integrable structure emerges when the $q$-parameters $q_1,q_2$ take special values. The second integrable structure is a $q$-difference analogue of the 1D Toda equation. The partition function satisfies this $q$-difference equation with respect to $Q$. Unlike the bigraded Toda hierarchy, this integrable structure exists for any values of $q_1,q_2$.

💡 Research Summary

The paper investigates a two‑parameter deformation of the melting‑crystal model, a statistical system equivalent to random plane partitions (or 3‑dimensional Young diagrams). By introducing two independent deformation parameters (q_{1}) and (q_{2}) the authors assign distinct weights to the three orthogonal directions of the crystal lattice. In addition to these anisotropic (q)‑weights, an infinite family of external potentials depending on the shape of the diagonal slice of a plane partition is added; the potentials are controlled by an infinite set of coupling constants (t_{1},t_{2},\dots). The resulting Boltzmann weight for a configuration (\pi) reads schematically as
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