We study the effects of primordial non-Gaussianity on the large-scale structure in the excursion set approach, accounting for correlations between steps of the random walks in the smoothed initial density field. These correlations are induced by realistic smoothing filters (as opposed to a filter that is sharp in k-space), but have been ignored by many analyses to date. We present analytical arguments – building on existing results for Gaussian initial conditions – which suggest that the effect of the filter at large smoothing scales is remarkably simple, and is in fact identical to what happens in the Gaussian case: the non-Gaussian walks behave as if they were smooth and deterministic, or ‘completely correlated’. As a result, the first crossing distribution (which determines e.g. halo abundances) follows from the single-scale statistics of the non-Gaussian density field – the so-called ‘cloud-in-cloud’ problem does not exist for completely correlated walks. Also, the answer from single-scale statistics is simply one half that for sharp-k walks.
We explicitly test these arguments using Monte Carlo simulations of non-Gaussian walks, showing that the resulting first crossing distributions, and in particular the factor 1/2 argument, are remarkably insensitive to variations in the power spectrum and the defining non-Gaussian process. We also use our Monte Carlo walks to test some of the existing prescriptions for the non-Gaussian first crossing distribution. Since the factor 1/2 holds for both Gaussian and non-Gaussian initial conditions, it provides a theoretical motivation (the first, to our knowledge) for the common practice of analytically prescribing a ratio of non-Gaussian to Gaussian halo abundances.