On the emptiness formation probability in the free-fermion six-vertex model with domain wall boundary conditions

Various representations are derived for the emptiness formation probability (a nonlocal correlation function describing the probability of a ferroelectric order) in the six-vertex model with domain wall boundary conditions in the case of weights satisfying the free-fermion condition. Starting from the known representation in terms of a multiple integral, the emptiness formation probability is expressed in terms of Hankel determinants and Fredholm ones. The nonlinear differential equations for this correlation function are also obtained. In particular, among these equations are those for the tau-functions of Toda chains, both for the finite and the semi-infinite ones.