A random countable set of points on a measure space such that the number of points in disjoint measurable subsets are independent and each count is Poisson-distributed with mean given by an intensity measure; the process is characterized by its intensity.

Principle

Complete spatial randomness: independence of counts on disjoint sets together with Poisson statistics for each set, determined by an intensity measure that may be constant (homogeneous) or spatially varying (inhomogeneous).

Demonstration

On R^d with Lebesgue measure and constant intensity λ>0, the homogeneous Poisson point process has for any bounded B the probability P(N(B)=k)=exp(-λ|B|)(λ|B|)^k/k! and points uniformly distributed in B conditional on N(B)=k.

Misapplication

Using a Poisson point process to model phenomena with strong repulsion or clustering (e.g., electrons with Coulomb repulsion), thereby ignoring interactions that violate independence of disjoint sets.

Consequence

Admits simple functional characterizations (Laplace functional, void probabilities), thinning and superposition rules, and serves as the null model for spatial randomness against which interaction models are tested.

Reversal

A determinantal or Gibbs point process encodes interaction (repulsion or attraction) between points; reversing the Poisson assumption replaces independence by specified correlation structure.

Boundary

Valid for phenomena where counts on disjoint sets are approximately independent and the intensity measure is σ-finite; excludes processes with fixed total count (binomial process) or those defined only via conditional intensities depending on past points.

Semantic Tension

Tension with the binomial point process: both put points in a region, but the binomial process conditions on a fixed number of points while the Poisson process has random counts with Poisson law.

Synthesis

A Poisson point process is the canonical model of complete randomness on a measure space: independent Poisson counts on disjoint sets with distribution determined by an intensity measure.