Two-point phase correlation function ($p_{ij}(x,x^{\prime })=\mathbb{E}[{\chi }_i(x){\chi }_j(x^{\prime })]$)

The indicator correlation gives the probability of specified phases at two points. Statistical homogeneity makes it depend on separation; isotropy makes it depend only on its length. In particular $p_{11}(0)=p_1$, so the connected correlation at zero separation is $p_1-p_1^2=p_1{p}_2$.