A new representation for the partition function of the six vertex model with domain wall boundaries

We obtain a new representation for the partition function of the six vertex model with domain wall boundaries using a functional equation recently derived by the author. This new representation is given in terms of a sum over the permutation group where the partial homogeneous limit can be taken trivially. We also show by construction that this partition function satisfies a linear partial differential equation.

💡 Research Summary

The paper addresses the long‑standing problem of finding a tractable expression for the partition function Z_N of the six‑vertex model with domain‑wall boundary conditions (DWBC). While the celebrated Izergin‑Korepin determinant gives an exact formula, its practical use is hampered by the difficulty of taking homogeneous limits and by the rapid growth of computational complexity for large lattice sizes. The author builds on a functional equation derived in earlier work, which encodes the invariance of Z_N under the exchange of spectral parameters λ_i and μ_j and reflects the underlying Yang‑Baxter symmetry of the R‑matrix.

The central result is a new representation of Z_N as a sum over the permutation group S_N. For each permutation σ the contribution is a product of simple rational functions f(λ_i, μ_{σ(i)}) and a pairwise interaction factor g(λ_i, λ_j; μ_{σ(i)}, μ_{σ(j)}). Explicitly, \