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Embed the set {1,...,n} into the unit interval I=[0,1] by θ(i) ≡ i/n. Then look at θ(A) for the fixed points A ⊂ {1,..,n} of a random permutation. Increasing n to infinity, the distribution of θ(A) should converge weakly to the Poisson point process on I with intensity being the standard Lebesgue measure.

Embed the set $\{1,...,n\}$ into the unit interval $I=[0,1]$ by $\theta(i)=\frac in$. Then look at $\theta(A)$ for the fixed points $A\subset \{1,..,n\}$ of a random permutation. Increasing n to infinity, the distribution of $\theta(A)$ should converge weakly to the Poisson point process on I with intensity being the standard Lebesgue measure.

Embed the set {1,...,n} into the unit interval I=[0,1] by θ(i) ≡ i/n. Then look at θ(A) for the fixed points A ⊂ {1,..,n} of a random permutation. Increasing n to infinity, the distribution of θ(A) should converge weakly to the Poisson random measure on I with intensity being the standard Lebesgue measure.

Embed the set {1,...,n} into the unit interval I=[0,1] by θ(i) ≡ i/n. Then look at θ(A) for the fixed points A ⊂ {1,..,n} of a random permutation. Increasing n to infinity, the distribution of θ(A) should converge weakly to the Poisson point process on I with intensity being the standard Lebesgue measure.

I used the unit interval as the space in which to embed the finite sets {1,...,n} but, really, any finite measure space with no atoms will do.

I used the unit interval as the space in which to embed the finite sets {1,...,n} but, really, any finite measure space (E,ℰ,μ) with no atoms will do. Just embed the finite subsets uniformly over the space (i.e., at random).