Evaluation of integrals for the emptiness formation probability in the square-ice model

TL;DR

This paper investigates the emptiness formation probability in the six-vertex model with domain wall boundary conditions, proposing a conjecture that relates its integral representation to a finite-size matrix determinant, and connecting it to the GUE Tracy-Widom distribution near the arctic curve.

Contribution

It introduces a conjecture linking the multiple integral representation of EFP to a finite-size Fredholm determinant, providing new insights into its structure and asymptotic behavior.

Findings

Conjecture relating MIR to Fredholm determinant at the ice point

Explicit evaluation of MIR for specific geometric parameters

Asymptotic connection to GUE Tracy-Widom distribution near the arctic curve

Abstract

We study the emptiness formation probability (EFP) in the six-vertex model with domain wall boundary conditions. We present a conjecture according to which at the ice point, i.e., when all the Boltzmann weights are equal, the known multiple integral representation (MIR) for the EFP can be given as a finite-size matrix determinant of Fredholm type. Our conjecture is based on the explicit evaluation of the MIR for particular values of geometric parameters and on two kinds of identities for the boundary correlation function. The obtained representation can be further written as the Fredholm determinant of some linear integral operator. We show that as the geometric parameters of the EFP are tuned to the vicinity of the arctic curve arising in the scaling limit, the conjectured determinant turns into the GUE Tracy--Widom distribution.