Talk:Correlation function

Different meaning for point processes

In the context of point processes, the term correlation function has a different meaning. There the k-point correlation function is more or less what you get by keeping $k$ points as variables and integrating out all other points. Say for a point process on space ${\mathcal {X}}$ which always yields n points, with (symmetric) probability density $\rho (x_{1},\ldots ,x_{n})$, for $k\leq n$ the k-point correlation function is

$R_{k}(x_{1},\ldots ,x_{k})={\frac {n!}{(n-k)!}}\int _{\mathcal {X}}dx_{k+1}\int _{\mathcal {X}}dx_{k+2}\ldots \int _{\mathcal {X}}dx_{n}\rho (x_{1},\ldots ,x_{n})$.

Note that the correlation function is is not zero if the points are independent (a Poisson process), but factorizes in such a case. A related concept is known as cluster function - essentially the correlation function minus what is implied by clusters of size smaller than k. So the higher cluster functions are zero for independent points. See for instance here: https://arxiv.org/pdf/solv-int/9804004 (equation (2.1), though they seem to have the wrong sign or a different convention, I think it should be $(-1)^{m-1}$). Perhaps a better reference is Madan Lal Mehta's textbook on random matrices. Icek~enwiki (talk) 17:44, 31 October 2026 (UTC) Edit: The sign is alright, it is alternating in the "single cluster" term, so this cluster function is, expressed in correlation functions as above,

$T_{n}(x_{1},\ldots ,x_{k})=\sum _{m=1}^{k}(-1)^{k-m}(m-1)!\sum _{G{\text{ set partition of }}\{1,\ldots ,k\}}\prod _{j=1}^{m}R_{|G_{j}|}(x_{G_{j,1}},\ldots ,x_{G_{j,|G_{j}|}})$

See equations (6.1.2) and (6.1.3) in Mehta, Madan Lal (2004). Random Matrices. Oxford: Elsevier. ISBN 0-12-088409-7. Icek~enwiki (talk) 14:12, 1 November 2026 (UTC)