New Limits for Large N Matrix and Tensor Models: Large D, Melons and Applications

Large N matrix models play an important role in modern theoretical physics, ranging from quantum chromodynamics to string theory and holography. However, they remain a difficult technical challenge because in most cases it is not known how to perform the sum over planar graphs, which dominate the models at large N. In this thesis, we study large D matrix models as they provide a framework to build new limits for matrix models in which the sum over planar graphs simplifies when D is large. The basic degrees of freedom are real matrices of size NxN with r additional indices of range D. These matrices can be interpreted as a real tensor of rank R=r+2 with indices of different ranges, making a compelling connection with tensor models. We define a new large D limit for the sum over Feynman graphs of fixed genus in matrix models, based on an enhanced large D scaling of the coupling constants. Using the combinatorial techniques developed in tensor models, we show that the resulting large D expansion is well-defined and organized according to a half-integer called the index. When N=D, the result also provides a new large N limit for general tensor models. In the large D limit, the sum over planar graphs of large D matrix models simplifies to a non-trivial sum over generalized melonic graphs. This class of graphs extends the one obtained in tensor models with standard scaling and allows for a wider class of interactions, including all the maximally single-trace terms. The general classification of generalized melonic graphs remains an open problem. However, in the case of the complete interaction of order R+1 for R a prime number, we identify them in detail and demonstrate that they exhibit the same important black hole-like features as the SYK model with q=(R+1)-fold random interactions, including the emergent conformal symmetry in the infrared regime and maximal chaos. The advantage of large D matrix models over the SYK model and its variants is that they correspond to genuine quantum field theories. In addition, for r=1, they have a natural interpretation in terms of D-brane constructions in string theory, making a possible relation with holography clearer. Another part of this thesis applies the tools developed in tensor models to study non-linear resonant flows in many variables. By averaging over both the tensor coupling and the initial conditions, we prove that in some regime of perturbation theory, melonic graphs dominate the dynamics and are responsible for turbulent energy cascades.Doctorat en Sciences