# Integrable structure of melting crystal model with two q-parameters - Mathematical Physics

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Abstract: This paper explores integrable structures of a generalized melting crystalmodel that has two $q$-parameters $q 1,q 2$. This model, like the ordinary onewith a single $q$-parameter, is formulated as a model of random planepartitions or, equivalently, random 3D Young diagrams. The Boltzmann weightcontains an infinite number of external potentials that depend on the shape ofthe diagonal slice of plane partitions. The partition function is thereby afunction of an infinite number of coupling constants $t 1,t 2,

.$ and an extraone $Q$. There is a compact expression of this partition function in thelanguage of a 2D complex free fermion system, from which one can see thepresence of a quantum torus algebra behind this model. The partition functionturns out to be a tau function times a simple factor of two integrablestructures simultaneously. The first integrable structure is the bigraded Todahierarchy, which determine the dependence on $t 1,t 2,

.$. This integrablestructure emerges when the $q$-parameters $q 1,q 2$ take special values. Thesecond integrable structure is a $q$-difference analogue of the 1D Todaequation. The partition function satisfies this $q$-difference equation withrespect to $Q$. Unlike the bigraded Toda hierarchy, this integrable structureexists for any values of $q 1,q 2$.

Author: ** Kanehisa Takasaki**

Source: https://arxiv.org/