Detection of spatial clustering in the 1000 richest SDSS DR8 redMaPPer
  clusters with Nearest Neighbor distributions
  Distances to the $k$-nearest-neighbor ($k$NN) data points from volume-filling
query points are a sensitive probe of spatial clustering. Here we present the
first application of $k$NN summary statistics to observational clustering
measurement, using the 1000 richest redMaPPer clusters ($0.1\leqslant
z\leqslant 0.3$) from the SDSS DR8 catalog. A clustering signal is defined as a
difference in the cumulative distribution functions (CDFs) or counts-in-cells
functions (CICs) of $k$NN distances from fixed query points to the observed
clusters versus a set of unclustered random points. We find that the $k=1,2$-NN
CDFs (and CICs) of redMaPPer deviate significantly from the randoms' across
scales of 35 to 155 Mpc, which is a robust signature of clustering. In addition
to $k$NN, we also measure the two-point correlation function for the same set
of redMaPPer clusters versus random points, which shows a noisier and less
significant clustering signal within the same radial scales. Quantitatively,
the $\chi^2$ distribution for both the $k$NN-CDFs and the two-point correlation
function measured on the randoms peak at $\chi^2\sim 50$ (null hypothesis),
whereas the $k$NN-CDFs ($\chi^2\sim 300$, $p = 1.54\times 10^{-36}$) pick up a
much more significant clustering signal than the two-point function
($\chi^2\sim 100$, $p = 1.16\times 10^{-6}$) when measured on redMaPPer.
Finally, the measured 3NN and 4NN CDFs deviate significantly from the predicted
$k=3, 4$-NN CDFs assuming an ideal Gaussian field, indicating that redMaPPer
clusters trace a non-Gaussian density field which is sensitively picked up by
$k$NN summary statistics. Therefore, the $k$NN method serves as a more
sensitive probe of clustering complementary to the two point correlation
function in sparse (beyond-Gaussian) observational data sets at large scales,
providing a novel approach for constraining cosmology and galaxy-halo
connection.
