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Summary: Notes on the Brydges-Kennedy-Abdesselam-Rivasseau
forest interpolation formula
There are many instances in mathmatical physics where one tries to under-
stand joint probability measures for a collection of random variables X1, . . . , XN ,
with N large, of the form
e
-
NP
i=1
V (xi)
dµC(x)
where dµC is a Gaussian measure on RN . The dependence between these ran-
dom variables is entierly due to the Gaussian measure which, in general, is given
by covariances
Cij = cov(Xi, Xj)
which do not vanish for i = j. A typical procedure one uses in this type of
problem is to try to interpolate between the given covariance matrix C and the
covariance obtained by killing the off-diagonal entries. The outcome is what is
called a cluster expansion in the constructive field theory literature. The first
such expansions appeared in the context of the construction of the 4 model in
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