##  [Poisson Point Process](/poisson-point-process-0) 

 Definition

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

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

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

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

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

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

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

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

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.