  We study the effect of non-Gaussian average over the random couplings in a
complex version of the celebrated Sachdev-Ye-Kitaev (SYK) model. Using a
Polchinski-like equation and random tensor Gaussian universality, we show that
the effect of this non-Gaussian averaging leads to a modification of the
variance of the Gaussian distribution of couplings at leading order in N. We
then derive the form of the effective action to all orders. An explicit
computation of the modification of the variance in the case of a quartic
perturbation is performed for both the complex SYK model mentioned above and
the SYK generalization proposed in D. Gross and V. Rosenhaus, JHEP 1702 (2017)
093.
