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).
If X is a random variable, the moments sn and cumulants (same as the Ursell functions) un are functions of X related by the exponential formula: (where
[1] The Ursell functions of a single random variable X are obtained from these by setting X = X1 = … = Xn.
The first few are given by Percus (1975) 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 Xi can be divided into two nonempty independent sets.