Some explicit results for the generalized emptiness formation probability of the six-vertex model

We study a multi-point correlation function of the six-vertex model on the square lattice with the domain wall boundary conditions which is called the generalized emptiness formation probability. This function describes probability of observing the ferroelectric order around all the vertices of any Ferrer diagram $\lambda$ at the top-left corner of the lattice. For the free-fermion model we derive and compare explicit formulas for this correlation function for two cases of diagram $\lambda$: the square and triangle. We found a connection of our formulas with the $\tau$-function of the sixth Painlevé equation.