the normalisation of our results. We define a re-normalised cross correlation $\xi^\prime$:$$\begin{aligned}
\xi^\prime=C \xi & = & \frac{C}{\overline{\rho}_1 (\overline{\rho}_2 + C)} \left< (\rho_1 - \overline{\rho}_1)\left(\rho^\prime_2 +C - (\overline{\rho^\prime_2 }+ C)\right)\right> \nonumber \\
  &= & \frac{1}{\overline{\rho}_1} \left< (\rho_1 - \overline{\rho}_1) (\rho^\prime_2) \right>\end{aligned}$$ By construction, Eq. (3) gives the correlation between the QSO number density field and the FIR surface brightness field (with a normalisation constant). It is also equivalent to the stacked SPIRE map signal at the positions of the QSOs minus the mean signal of the map (which is equal to zero by design). This re-normalised cross-correlation, equivalent to stacking, is no longer dimensionless, but has the same unit as the map.

To estimate the two-point angular cross-correlation function $\xi$ between the quasar samples and the SPIRE maps, we adapt the Landy & Szalay (1993) auto-correlation estimator[^10], $$w^{\rm QS}(\theta) = \frac{D_1D_2(\theta) - D_1R_2(\theta) - R_1D_2(\theta) + R_1R_2(\theta)}{R_1R_2(\theta)}.$$ Here $D_1$ represents the pair-counts in the real quasar sample, $D_2$ the CIB at a given wavelength, $R_1$ the simulated quasar sample with the same angular selection effects as the real quasar sample, and $R_2$ the simulated CIB with the same noise properties as the real SPIRE maps. $D_1D_2(\theta)$ represents the total emission 