  We consider functional-integral quantisation of the moduli of all quantum
metrics defined as square-lengths $a$ on the edges of a Lorentzian square
graph. We determine correlation functions and find a fixed relative uncertainty
$\Delta a/<a>= 1/\sqrt{8}$ for the edge square-lengths relative to their
expected value $<a>$. The expected value of the geometry is a rectangle where
parallel edges have the same square-length. We compare with the simpler theory
of a quantum scalar field on such a rectangular background. We also look at
quantum metric fluctuations relative to a rectangular background, a theory
which is finite and interpolates between the scalar theory and the fully
fluctuating theory.
