Nearest-Neighbor Graphs

Consider a set  of  points that are independently and uniformly distributed in the -dimensional unit cube. Let  ∈  . There exists almost-surely  ∈  such that  6=  and | − |  | − | for all  ∈  ,  6= ,  6= . The point  is called the nearest neighbor of  and we write  ≺ . Note that  ≺  does not imply  ≺ . Draw an edge connecting  and  if and only if  ≺ ; the resulting graph of  vertices and ≤  edges is called the nearest-neighbor graph  on  . What is the probability, (), given  ∈  , that  ≺  implies  ≺ ? Such a pair is isolated from the rest of , in the sense that the only edge touching  or  is the edge that connects  and . We have [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]