Random tensors
This book is a self-contained introductory text to the theory of random tensors. The book presents theory and its applications to physics. This book introduces a framework for studying random geometries in any dimensions. This book shows that many results in random matrices, most notably, Hooft's \(1/N\) expansion can be generalized to higher dimension.NEWLINENEWLINERandom tensor models generalize random matrix models and provide a framework for the study of random geometries. The development of matrix models is one of the most impressive achievements of modern theoretical physics. Generalizing random matrices, random tensors generate Feyman graphs, which can be interpreted as topological spaces.NEWLINENEWLINEThe book consists of three parts. The first part consists of Chapters 2 to 6. In this part, the general framework and the main results on random tensors are introduced. The second part consists of Chapters 7 to 11. In this part, specific examples of random tensor models are discussed. The third part consists of appendices related to Weingarten functions, probability measures, Borel summability and BKAR formula.NEWLINENEWLINEChapter 1 is introductory.NEWLINENEWLINEChapter 2: Preliminaries related to tensors, invariants, edge colored graphs, matrices, connected and disconnected trace invariants, uniqueness of the decomposition on trace invariants, invariant probability measures are discussed. Also representation of invariants as edge colored graphs is introduced.NEWLINENEWLINEChapter 3: In this chapter, the edge colored graphs are studied. It is shown that each graph is a \(D\)-complex dual to a vertex colored triangulation. Also, open graphs and their boundary graphs are discussed. The notion of ``degree'' associated with to a connected edge colored graph are introduced.NEWLINENEWLINEChapter 4: In this chapter, the classification of graphs at fixed degree is given. It is shown that for any fixed degree, one obtains an infinite but exponentially bounded family of graphs, whose generating function can be written explicitly. It is shown that the number of reduced schemes with a fixed (reduced) degree is finite.NEWLINENEWLINEChapter 5: Classification of graphs at fixed degree is discussed. The geometry of the infinite family of melons is studied. It is shown that, with a certain metric, this family reproduces the behavior of branched polymers. The spectral dimension of melons is found.NEWLINENEWLINEChapter 6: Random tensors are studied. It is shown that invariant tensor measures have a very strong universality property. Gaussian distribution for random tensors is discussed. In particular, its moments are computed. It is proved that any properly uniformly bounded trace invariant probability measure becomes Gaussian in the large \(N\) limit. This is the universality theorem for tensors.NEWLINENEWLINETheorem (the universality theorem): Consider \(N^D\) random variables \(\mathbb{T}_{\alpha^D}\) whose joint distribution is trace invarient with covarience \(K(\mathcal{B}^{(2)}, \mu_N)\) and is properly uniformly bounded. Then in the large \(N\) limit the tensor \(\mathbb{T}_{\alpha^D}\) converges in distribution to a Gaussian tensor of covariance \(K(\mathcal{B}^{(2)}) = \lim_{n \to \infty} K(\mathcal{B}^{(2)}, \mu_N)\).NEWLINENEWLINEChapter 7: In this chapter, a review of some features of matrix models is given. In particular review of the \(1/N\) expansion, the continuum limit, the double scaling limit and the Schwinger-Dyson equations is given.NEWLINENEWLINEChapter 8: In this chapter, the perterbative expansion of invariant tensor measures is studied. It is shown that under certain conditions, the moments of such measures admit a \(1/N\) expansion and that all such measures are properly uniformly bounded. The continuum limit of random tensors models and their Schwinger-Dyson equations are discussed.NEWLINENEWLINEChapter 9: In this chapter, the tensor model with an arbitrary quartic expansion is studied. It is shown that using the Hubbard-Stratonovich intermediate field representation, the quartic model can be reformulated in terms of a model of edge colored maps. Analyticity results for the partition function and the cumulants in the constructive sense are proved. It is shown that the \(1/N\) expansion and the proper uniform boundedness holds.NEWLINENEWLINEChapter 10: The double scaling limit of quartic melonic models is discussed. It is shown that the melonic family in these models can be analytically resummed and that this resummation is encoded in a translation of the intermediate matrix fields to a nontrivial vacuum. After this translation, one obtains explicitly generating function for the scheme as a path integral. As a consequence, it is shown that the continuum limit of tensor models corresponds to a phase transition (in the field theory sense) associated with a symmetry breaking. This intermediate field representation is used to discuss the double scaling limit of the quartic melonic model.NEWLINENEWLINEChapter 11: In this chapter, quartic tensor model with one interaction for a tensor of arbitrary rank is analyzed. It is shown that in this case the critical point corresponds to a phase transition in the tensor model to a breaking of the unitary symmetry. It is also shown that in a double scaling limit, the symmetric phase corresponds to a theory of infinitely refined random surfaces and the broken phase corresponds to a theory of infinitely refined random nodal surfaces. At leading order in the double scaling limit planar surfaces dominate in the symmetric phase and planar nodal surfaces dominate in the broken phase.NEWLINENEWLINEChapter 12: In this chapter, the interpretation of tensor models as generating Euclidean dynamical triangulations is discussed. Also some applications of tensor models as models of random geometry are discussed.
scientific article; zbMATH DE number 6638014