We prove an explicit formula for the law in zero of the solution of a class
of elliptic SPDE in $\mathbb{R}^2$. This formula is the simplest instance of
dimensional reduction, discovered in the physics literature by Parisi and
Sourlas (1979), which links the law of an elliptic SPDE in $d + 2$ dimension
with a Gibbs measure in $d$ dimensions. This phenomenon is similar to the
relation between an $\mathbb{R}^{d + 1}$ dimensional parabolic SPDE and its
$\mathbb{R}^d$ dimensional invariant measure. As such, dimensional reduction of
elliptic SPDEs can be considered a sort of elliptic stochastic quantisation
procedure in the sense of Nelson (1966) and Parisi and Wu (1981). Our proof
uses in a fundamental way the representation of the law of the SPDE as a
supersymmetric quantum field theory. Dimensional reduction for the
supersymmetric theory was already established by Klein et al. (1984). We fix a
subtle gap in their proof and also complete the dimensional reduction picture
by providing the link between the elliptic SPDE and the supersymmetric model.
Even in our $d = 0$ context the arguments are non-trivial and a
non-supersymmetric, elementary proof seems only to be available in the Gaussian
case.