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The use of summary statistics beyond the two-point correlation function toanalyze the non-Gaussian clustering on small scales is an active field ofresearch in cosmology. In this paper, we explore a set of new summarystatistics -- the $k$-Nearest Neighbor Cumulative Distribution Functions($k{\rm NN}$-${\rm CDF}$). This is the empirical cumulative distributionfunction of distances from a set of volume-filling, Poisson distributed randompoints to the $k$-nearest data points, and is sensitive to all connected$N$-point correlations in the data. The $k{\rm NN}$-${\rm CDF}$ can be used tomeasure counts in cell, void probability distributions and higher $N$-pointcorrelation functions, all using the same formalism exploiting fast searcheswith spatial tree data structures. We demonstrate how it can be computedefficiently from various data sets - both discrete points, and thegeneralization for continuous fields. We use data from a large suite of$N$-body simulations to explore the sensitivity of this new statistic tovarious cosmological parameters, compared to the two-point correlationfunction, while using the same range of scales. We demonstrate that the use of$k{\rm NN}$-${\rm CDF}$ improves the constraints on the cosmological parametersby more than a factor of $2$ when applied to the clustering of dark matter inthe range of scales between $10h^{-1}{\rm Mpc}$ and $40h^{-1}{\rm Mpc}$. Wealso show that relative improvement is even greater when applied on the samescales to the clustering of halos in the simulations at a fixed number density,both in real space, as well as in redshift space. Since the $k{\rm NN}$-${\rmCDF}$ are sensitive to all higher order connected correlation functions in thedata, the gains over traditional two-point analyses are expected to grow asprogressively smaller scales are included in the analysis of cosmological data.

Nearest Neighbor distributions: new statistical measures for cosmological clustering