  We present a construction of the integrand of the correlation function of
four stress-tensor multiplets in N=4 SYM at weak coupling. It does not rely on
Feynman diagrams and makes use of the recently discovered symmetry of the
integrand under permutations of external and integration points. This symmetry
holds for any gauge group, so it can be used to predict the integrand both in
the planar and non-planar sectors. We demonstrate the great efficiency of
graph-theoretical tools in the systematic study of the possible permutation
symmetric integrands. We formulate a general ansatz for the correlation
function as a linear combination of all relevant graph topologies, with
arbitrary coefficients. Powerful restrictions on the coefficients come from the
analysis of the logarithmic divergences of the correlation function in two
singular regimes: Euclidean short-distance and Minkowski light-cone limits. We
demonstrate that the planar integrand is completely fixed by the procedure up
to six loops and probably beyond. In the non-planar sector, we show the absence
of non-planar corrections at three loops and we reduce the freedom at four
loops to just four constants. Finally, the correlation function/amplitude
duality allows us to show the complete agreement of our results with the
four-particle planar amplitude in N=4 SYM.
