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tokens: it may not be the case for () point statistics. For instance, peak-peak correlation functions on small scales are not very well reproduced by this approximation even for large contrasts because the contribution from the other critical points actually dominates at small distance; there is at least one saddle point between two peaks. However, in the context of this work, we impose the peak constraint in one location only and therefore the rare peak approximation is expected to be accurate for . As an illustration, Fig. 2 shows the Gaussian mean number density of minima, saddle points, and peaks ( Longuet-Higgins 1957 ; Adler 1981 , and later generalised to weakly non-Gaussian fields by Pogosyan et al. 2011 ) and compares the latter to the high- approximation (related to the genus) which can be easily computed The relative error between the number density of peaks and its high- approximation is shown on the right-hand panel of Fig. 2 . Fig. 4 Zero lag annulus correlation function in units of for a central peak with height and eigenvalues and in for a Gaussian random field with power spectrum smoothed with a Gaussian filter (similar to Fig. 3 when the two points on the annulus are the same). The separation is given in units of the smoothing length. 2.4. Density correlations on the circle surrounding a central peak with given geometry The expected product of projected density in two locations of space and such that and and given a peak in of height and curvatures along the first and second coordinates can be analytically computed. For instance for a power-law power spectrum with (and ), we get where is the trace of the density Hessian at the location of the peak, is the ellipticity of the peak and the unconstrained correlation function is To start with, Fig. 4 shows the zero lag contribution to the annulus correlation function (i.e. when the two points are at the same location on the annulus, ) in the frame of the central peak. As expected the amplitude of fluctuations around the peak have an ellipsoidal shape that is more elongated along the smallest curvature . This zero-lag annulus correlation is dominated by the square of the mean density profile at small separations and by the fluctuations at larger separations. Figure 3 then shows the full constrained correlation function on the annulus. We use different orientations of the pair () with regard to the axis of smaller curvature (corresponding to ) of the central peak. The orientation of is described by the angle which is taken to be 0, and from the left-hand to the right-hand panel. On each plot, the angle between and , namely , vary between 0 and and the separation to the central peak is described by the value . We show the result for two different peak heights: the most likely value (top panels) and a rarer case (bottom panels), which is more relevant to our study. In each case respectively, we fix the peak curvatures to their most likely values and , respectively (see Appendix A for a