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

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 $t_1,\dots ,t_k\in \{1,\dots ,6\}$ label the six possible vertex types, and $\{a_t:1\le t\le 6\}$ are the corresponding six-vertex weights. For each $1\le p\le k$, the four points $x^{2p-1},x^{2p},y^{2p-1},y^{2p}\in (\mathbb{Z}\mathrm{/}2{)}^2$ 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}{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 $h(c(x),c(y);w_1,w_2)$ 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.