  A sizeable level of non-Gaussianity in the primordial cosmological
perturbations may be induced by a large trispectrum, i.e. by a large connected
four-point correlation function. We compute the effect of a primordial
non-Gaussian trispectrum on the halo mass function, within excursion set
theory. We use the formalism that we have developed in a previous series of
papers and which allows us to take into account the fact that, in the presence
of non-Gaussianity, the stochastic evolution of the smoothed density field, as
a function of the smoothing scale, is non-markovian. In the large mass limit,
the leading-order term that we find agrees with the leading-order term of the
results found in the literature using a more heuristic Press-Schecther
(PS)-type approach. Our approach however also allows us to evaluate
consistently the subleading terms, which depend not only on the four-point
cumulant but also on derivatives of the four-point correlator, and which cannot
be obtained within non-Gaussian extensions of PS theory. We perform explicitly
the computation up to next-to-leading order.
