  We derive an analytical expression for a novel large-scale structure
observable: the line correlation function. The line correlation function, which
is constructed from the three-point correlation function of the phase of the
density field, is a robust statistical measure allowing the extraction of
information in the non-linear and non-Gaussian regime. We show that, in
perturbation theory, the line correlation is sensitive to the coupling kernel
F_2, which governs the non-linear gravitational evolution of the density field.
We compare our analytical expression with results from numerical simulations
and find a 1-sigma agreement for separations r<30 Mpc/h. Fitting formulae for
the power spectrum and the non-linear coupling kernel at small scales allow us
to extend our prediction into the strongly non-linear regime where we find a
1-sigma agreement with the simulations for r<2 Mpc/h. We discuss the advantages
of the line correlation relative to standard statistical measures like the
bispectrum. Unlike the latter, the line correlation is independent of the bias,
in the regime where the bias is local and linear. Furthermore, the variance of
the line correlation is independent of the Gaussian variance on the modulus of
the density field. This suggests that the line correlation can probe more
precisely the non-linear regime of gravity, with less contamination from the
power spectrum variance.
