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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, which 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 $Ntimes N$ with $r$ additional indices of range $D$. They 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. Then, 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 provides a new large $N$ limit for general $text{O}(N)^R$ invariant tensor models. In the large $D$ limit, the sum over planar graphs of large $N$ 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 features as the SYK model with $q=(R+1)$-fold random interactions, including the emergent conformal symmetry in the infrared regime and maximal chaos.
We investigate the existence and properties of a double asymptotic expansion in $1/N^{2}$ and $1/sqrt{D}$ in $mathrm{U}(N)timesmathrm{O}(D)$ invariant Hermitian multi-matrix models, where the $Ntimes N$ matrices transform in the vector representation
In this work we revisit the problem of solving multi-matrix systems through numerical large $N$ methods. The framework is a collective, loop space representation which provides a constrained optimization problem, addressed through master-field minimi
We construct a class of matrix models, where supersymmetry (SUSY) is spontaneously broken at the matrix size $N$ infinite. The models are obtained by dimensional reduction of matrix-valued SUSY quantum mechanics. The potential of the models is slowly
We study a supersymmetric tensor model with four supercharges and $O(N)^3$ global symmetry. The model is based on a chiral scalar superfield with three indices and quartic tetrahedral interaction in the superpotential, which is relevant below three d
Noncompact SO(1,N) sigma-models are studied in terms of their large N expansion in a lattice formulation in dimensions d geq 2. Explicit results for the spin and current two-point functions as well as for the Binder cumulant are presented to next to