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The causal dynamical triangulations approach aims to construct a quantum theory of gravity as the continuum limit of a lattice-regularized model of dynamical geometry. A renormalization group scheme--in concert with finite size scaling…
In the usual matrix-model approach to random discretized two-dimensional manifolds, one introduces n Ising spins on each cell, i.e. a discrete version of 2D quantum gravity coupled to matter with a central charge n/2. The matrix-model…
The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing for the safe use of perturbation theory…
Of theories for why large-scale machine learning models generalize despite being vastly overparameterized, which of their assumptions are needed to capture the qualitative phenomena of generalization in the real world? On one hand, we find…
We consider random Hermitian matrices made of complex or real $M\times N$ rectangular blocks, where the blocks are drawn from various ensembles. These matrices have $N$ pairs of opposite real nonvanishing eigenvalues, as well as $M-N$ zero…
The normal form theory for polynomial vector fields is extended to those for $C^\infty$ vector fields vanishing at the origin. Explicit formulas for the $C^\infty$ normal form and the near identity transformation which brings a vector field…
A method of ``blocking'' triangulations that rests on the self-similarity feature of dynamically triangulated random manifolds is proposed. The method is used to define the renormalization group for random geometries. As an illustration,…
We consider generalized one-matrix models in which external fields allow control over the coordination numbers on both the original and dual lattices. We rederive in a simple fashion a character expansion formula for these models originally…
The results of the renormalization group are commonly advertised as the existence of power law singularities near critical points. The classic predictions are often violated and logarithmic and exponential corrections are treated on a…
Matrix models are a highly successful framework for the analytic study of random two dimensional surfaces with applications to quantum gravity in two dimensions, string theory, conformal field theory, statistical physics in random geometry,…
An exact renormalization group equation is derived for the free energy of matrix models. The renormalization group equation turns out to be nonlinear for matrix models, as opposed to linear for vector models. An algorithm for determining…
We introduce a class of variational states to describe quantum many-body systems. This class generalizes matrix product states which underly the density-matrix renormalization group approach by combining them with weighted graph states.…
These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general…
In these lecture notes prepared for the 11th Taiwan Spring School, Taipei 1997}, and updated for the Saalburg summer school 1998, we review the solutions of O(N) or U(N) models in the large N limit and as 1/N expansions, in the case of…
We provide a non-technical overview of recent extensions of renormalization methods and techniques to Group Field Theories (GFTs), a class of combinatorially non-local quantum field theories which generalize matrix models to dimension $d…
Lattice regularization is a standard technique for the nonperturbative definition of a quantum theory of fields. Several approaches to the construction of a quantum theory of gravity adopt this technique either explicitly or implicitly. A…
We review the recent exact solution of a matrix model which interpolates between flat and random lattices. The importance of the results is twofold: Firstly, we have developed a new large N technique capable of treating a class of matrix…
We study critical and universal behaviors of unitary invariant non-gaussian random matrix ensembles within the framework of the large-N renormalization group. For a simple double-well model we find an unstable fixed point and a stable…
We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the…
As applied to quantum theories, the program of renormalization is successful for `renormalizable models' but fails for `nonrenormalizable models'. After some conceptual discussion and analysis, an enhanced program of renormalization is…