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 $k$-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 $t_1,\ldots,t_k \in \{1,\ldots,6\}$ label the six possible vertex types, and $\{a_t : 1 \leq t \leq 6\}$ are the corresponding six-vertex weights. For each $1 \leq p \leq k$, the four points $x^{2p-1}, x^{2p}, y^{2p-1}, y^{2p} \in (\mathbb{Z}/2)^2$ are $t_p$-dependent choices among the midpoints of the edges incident to $v^p$. The correlation kernel $L$ has the contour integral representation \begin{align*} L(x,y) = \oint_{|w_1|=1} \oint_{|w_2|=1} \frac{dw_1}{2\pi i\, w_1}\, \frac{dw_2}{2\pi i\, 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 $h\big(c(x),c(y);w_1,w_2\big)$ is a simple rational function of $(w_1,w_2)$ that depends on $x$ and $y$ only through their orientations $c(x)$ and $c(y)$. Our proof is fully self-contained: we construct a determinantal point process on $\mathbb{Z}^2$ and identify the six-vertex model as its pushforward under an explicit mapping.