Some Explicit Results for the Generalized Emptiness Formation Probability of the Six-Vertex Model

Abstract

We study a multi-point correlation function of the six-vertex model on the square lattice with domain wall boundary conditions which is called the generalized emptiness formation probability. This function describes the probability of observing ferroelectric order around all vertices of any Ferrers diagram λ at the top left corner of the lattice. For the free fermion model, we derive and compare explicit formulas for this correlation function in two cases: when the diagram λ has a square or a triangular shape. We find a connection between our formulas and the τ-function of the sixth Painlevé equation. Bibliography: 25 titles.