  In the standard model for structure formation, bound objects originate from
the gravitational collapse of small perturbations arising from quantum
fluctuations with random phases. In other scenarios, based on defects,
structures are seeded by localized energy density. In principle, it is possible
to differentiate between these models on the basis of their statistical
properties; only in the former case is the initial density field an
almost-perfect random gaussian field. In this paper, we investigate the use of
the trispectrum of the galaxy density field, which is the connected four-point
function in Fourier space, as a discriminant between gaussian and non-gaussian
models. It has the advantage of having only weak non-linear growth. We define a
related statistic $\tau$ which, as a test of the gaussian hypothesis, is
independent of cosmology, the power spectrum and biasing, in real space, and
which is, in principle, a measure of the departure from gaussian statistics.
For galaxy redshift surveys, the statistic depends on cosmology and bias only
through the potentially observable parameter $\beta$. We compute the expected
errors on the estimate of $\tau$, and demonstrate with numerical simulations
that it can be a useful discriminant of models, with the important proviso that
any bias is linear on large scales. Whether it is the most effective method is
uncertain and depends on the nature of the departure from gaussianity.
