In this chapter, we first review the Sachdev–Ye–Kitaev (SYK) model, which is a quantum mechanical model of N fermions. The model is a quenched model, which means that the coupling constant is a random tensor with Gaussian distribution. The SYK model is dominated in the large N limit by melonic graphs, in the same way the tensor models presented in the previous three chapters are dominated by melonic graphs. We then present a purely graph theoretical proof of the melonic dominance of the SYK model. It is this property which led E. Witten to relate the SYK model to the coloured tensor model. In the rest of the chapter we deal with the so-called coloured SYK model, which is a particular case of the generalisation of the SYK model introduced by D. Gross and V. Rosenhaus. We first analyse in detail the leading order and next-to-leading order vacuum, two- and four-point Feynman graphs of this model. We then exhibit a thorough asymptotic combinatorial analysis of the Feynman graphs at an arbitrary order in the large N expansion. We end the chapter by an analysis of the effect of non-Gaussian distribution for the coupling of the model.
Next-to-Leading Order in the Large N Expansion of the Multi-Orientable Random Tensor Model
