Asymptotic evaluation of the doubly‑refined six‑vertex DWBC partition function

Determine the large‑n asymptotics of the partially inhomogeneous (doubly‑refined) six‑vertex model partition function with domain wall boundary conditions, Z_n(0^{n-1}, -ξ_1; t^{n-1}, t+ξ_2; γ), which the paper represents as a framed Hankel determinant, in order to characterize the joint distribution of the locations of the c‑type vertices in the first row and the last column of the n×n lattice.

Background

The six‑vertex model with domain wall boundary conditions (DWBC) admits an exact determinantal expression for its partition function via the Izergin–Korepin formula. In the homogeneous case, this reduces to a Hankel determinant, enabling rigorous asymptotics across all three phase regions using Riemann–Hilbert techniques.

A partially inhomogeneous ("refined") limit tracks boundary statistics, such as the location of c‑type vertices on the last column, yielding a bordered Hankel determinant whose asymptotics have been analyzed to extract Gaussian and geometric fluctuation regimes. A more involved "doubly‑refined" limit simultaneously tracks the c‑type vertex in both the last row and last column, leading to a framed Hankel determinant (equation \eqref{eq:6v-doubly_refined}).

While the bordered case has been addressed, the asymptotic analysis of the doubly‑refined (framed) expression as n→∞ remains open. Establishing this would provide information about the joint distribution of the two boundary c‑type vertex locations.

References

It is an open and interesting question to evaluate eq:6v-doubly_refined as n\to\infty in order to obtain information on the joint distribution of the location of the c-type vertices in the first row and column.

— Bordered and Framed Toeplitz and Hankel Determinants with Applications to Integrable Probability (2401.01971 - Gharakhloo et al., 3 Jan 2024) in End of Section 6 (The six-vertex model with DWBC), after equation \eqref{eq:6v-doubly_refined}