Explicit correlation functions for the six-vertex model in the free-fermion regime

Samuel G. G. Johnston; Rohan Shiatis

Paper

Explicit correlation functions for the six-vertex model in the free-fermion regime

Abstract

In this article, we show that, in the free-fermion regime of the six-vertex model, all

-point correlation functions of vertex types admit a determinantal representation: \begin{align*} \mathbb{P}\Bigg( \bigcap_{p=1}^k \{ \text{vertex at } v^p \text{ has type } t_p \} \Bigg) = \left( \prod_{p=1}^k a_{t_p} \right) \det\big[ L(x^i,y^j) \big]_{i,j=1}^{2k}, \end{align*} where

label the six possible vertex types, and

are the corresponding six-vertex weights. For each

, the four points

are

-dependent choices among the midpoints of the edges incident to

. The correlation kernel

has the contour integral representation \begin{align*} L(x,y) = \oint_{|w_1|=1} \oint_{|w_2|=1} \frac{dw_1}{2πi\, w_1}\, \frac{dw_2}{2πi\, w_2}\, w_1^{\,y_1 - x_1}\, w_2^{\,y_2 - x_2}\, h\big(c(x),c(y);w_1,w_2\big), \end{align*} where

is a simple rational function of

that depends on

and

only through their orientations

and

. Our proof is fully self-contained: we construct a determinantal point process on

and identify the six-vertex model as its pushforward under an explicit mapping.