Connected $(n,m)$-Point Functions of Diagonal $2$-BKP Tau-Functions, and Spin Double Hurwitz Numbers

We derive an explicit formula for the connected $(n,m)$-point functions associated to an arbitrary diagonal tau-function of the $2$-BKP hierarchy using computation of neutral fermions and boson-fermion correspondence of type $B$, and then apply this formula to the computation of connected spin double Hurwitz numbers. This is the type $B$ analogue of \cite{wy2}.