The K-th nearest neighbor random walk (Xₙ) ₍ ₀ on a homogeneous Poisson point process χ on ᵈ (d 1), starts at the origin and at each step picks its next Poisson point among its closest neighbors according to i. i. d. labels having the same distribution as K. Our main result (Theorem 1) states that the number of Poisson points visited by (Xₙ) ₍ ₀ admits an exponential decay whenever the random variable K has a bounded support (BS). In particular, the K-th nearest neighbor random walk visits finitely many Poisson points if and only if K satisfies Assumption (BS). To prove it, we introduce the key notion of pioneer point which allows us to deal with the region of ᵈ already explored by (Xₙ) ₍ ₀. Still under Assumption (BS), we also prove an exponential decay for the Euclidean length of the trajectory performed by (Xₙ) ₍ ₀ (Theorem 2). Finally, and quite surprisingly, we exhibit an example of label distribution with bounded support for which the K-th nearest neighbor random walk discovers new Poisson points after a number of steps whose tail distribution is at least polynomial (Theorem 3).
Basdevant et al. (Tue,) studied this question.