k-Nearest-Neighbours Gibbs Processes

#### E. Bertin, J.-M. Billiot, R. Drouilhet

1999, v.5, №2, 219-234

ABSTRACT

The present study introduces a class of directed $k$-nearest-neighbours point processes. The neighbourhood relation is non-symmetric and depends on the realization of the process. The existence of stationary Gibbs states is proved for such models in $R^m$. Furthermore, under a finite range condition, uniqueness of stationary Gibbs states is proved for sufficiently small intensity. Finally, original simulations are proposed using a direct adaptation to our class of models of the Geyer and Moller algorithm.

Keywords: stochastic geometry,Gibbs point processes,k-nearest-neighbours graph,equilibriumequations,correlation functions