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This is similar to asking for a given point what is the probability that another point lies in an interval of d less than it? This is proportional to (1-d/N)^(N-1) or equivalently exp((N-1)*log(1-d/N)).
For small x, log(1-x) ~= -x.
So the probability of d is ~prop exp(-d)
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The fact that you have N points and an interval of N explains why your scale parameter is 1, but it should hold for M != N points, just with a different lambda.
If $U_{(i)}$ and $U_{(j})$ represent the $i^{th}$ and $j^{th}$ order statistic respectively, then the joint distribution is given by
$$ f_{u,v}(U_{(i)}, U_{(j)}# $$