In statistical mechanics, an Ursell function or connected correlation function, is a cumulant of a random variable. It can often be obtained by summing over connected Feynman diagrams (the sum over all Feynman diagrams gives the correlation functions).
The Ursell function was named after Harold Ursell, who introduced it in 1927.
Definition.
If "X" is a random variable, the moments "s""n" and cumulants (same as the Ursell functions) "u""n" are functions of "X" related by the exponential formula:
 formula_1
(where formula_2 is the expectation).
The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.
formula_3
The Ursell functions of a single random variable "X" are obtained from these by setting "X" 
 "X"1 
 "X""n".
The first few are given by
formula_4
Characterization.
 showed that the Ursell functions, considered as multilinear functions of several random variables, are uniquely determined up to a constant by the fact that they vanish whenever the variables "X""i" can be divided into two nonempty independent sets.
References.
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