Emptiness formation probability in the domain-wall six-vertex model

Abstract

Abstract. The emptiness formation probability in the six-vertex model with domain wall boundary conditions is considered. This correlation function allows one to address the problem of limit shapes in the model. We apply the quantum inverse scattering method to calculate the emptiness formation probability for the inhomogeneous model. For the homogeneous model, the result is given both in terms of certain determinant and as a multiple integral representation.

The Limit Shape of Large Alternating Sign Matrices

“…As in [12,13], we call emptiness formation probability (EFP) and denote by F (r,s) N , where r, s = 1, . .…”

“…Formula (4) has been derived in [12], using the quantum inverse scattering method [16,17] and the results of papers [9,10,15]. This formula has also been discussed in [13] (see equation (3.6) therein).…”

“…, z s . This follows through the symmetrization procedure explained in [18], and some additional identity proven in [12]. We are interested in the behaviour of EFP in the so-called scaling limit, that is in the limit where r, s and N are all large, with the ratios r/N and s/N kept finite (and smaller than 1).…”

“…In paper [12] a multiple integral representation for EFP has been derived. Here, we study the multiple integral representation for large size of the lattice (corresponding to large size of ASMs).…”

Emptiness formation probability and Painlevé V equation in the XY spin chain

“…It is straightforward to see that for these states, the EFP is P = (1/2) L . Note that the same is obtained from the asymptotic expression (14), which in this case is exact since A(0, 1) = − log 2 and E(0, 1) = 0. This observation was extended in [21] to the ground state along the circle h 2 + γ 2 = 1.…”