scholarly journals Colored Triangulations of Arbitrary Dimensions are Stuffed Walsh Maps

10.37236/5614 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Valentin Bonzom ◽  
Luca Lionni ◽  
Vincent Rivasseau
Keyword(s):  
Matrix Models ◽  
Building Blocks ◽  
Field Method ◽  
Fixed Number ◽  
Colored Graphs ◽  
Intermediate Field ◽  
Tensor Models ◽  
Combinatorial Maps ◽  
Arbitrary Dimensions ◽  
Number Of Faces

Regular edge-colored graphs encode colored triangulations of pseudo-manifolds. Here we study families of edge-colored graphs built from a finite but arbitrary set of building blocks, which extend the notion of $p$-angulations to arbitrary dimensions. We prove the existence of a bijection between any such family and some colored combinatorial maps which we call stuffed Walsh maps. Those maps generalize Walsh's representation of hypermaps as bipartite maps, by replacing the vertices which correspond to hyperedges with non-properly-edge-colored maps. This shows the equivalence of tensor models with multi-trace, multi-matrix models by extending the intermediate field method perturbatively to any model. We further use the bijection to study the graphs which maximize the number of faces at fixed number of vertices and provide examples where the corresponding stuffed Walsh maps can be completely characterized.

A First Principles Approach for Partitioning Linear Response Properties into Additive and Cooperative Contributions

2018 ◽  
Author(s):  
Daniel Lambrecht ◽  
Eric Berquist
Keyword(s):  
First Principles ◽  
Linear Response ◽  
Building Blocks ◽  
Field Method ◽  
Response Property ◽  
Response Properties ◽  
Consistent Field ◽  
Molecular Building Blocks ◽  
Self Consistent ◽  
Self Consistent Field

We present a first principles approach for decomposing molecular linear response properties into orthogonal (additive) plus non-orthogonal/cooperative contributions. This approach enables one to 1) identify the contributions of molecular building blocks like functional groups or monomer units to a given response property and 2) quantify cooperativity between these contributions. In analogy to the self consistent field method for molecular interactions, SCF(MI), we term our approach LR(MI). The theory, implementation and pilot data are described in detail in the manuscript and supporting information.

From random matrices to random tensors

2021 ◽  
pp. 166-177
Author(s):  
Adrian Tanasa
Keyword(s):  
Matrix Models ◽  
Random Matrices ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Random Surface ◽  
The Third ◽  
Planar Surfaces ◽  
Tensor Models ◽  
Feynman Graphs ◽  
The Matrix

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).

scholarly journals Compact n-Manifolds via (n + 1)-Colored Graphs: a New Approach

2020 ◽  
Vol 27 (01) ◽  
pp. 95-120
Author(s):  
Luigi Grasselli ◽  
Michele Mulazzani
Keyword(s):  
Arbitrary Dimension ◽  
Fundamental Groups ◽  
New Approach ◽  
Colored Graphs ◽  
Connected Sums ◽  
Tensor Models ◽  
Computer Aided ◽  
The One

We introduce a representation via (n+1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is a generalization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three, and it is dual to the one given by Pezzana in the 1970s. In this context we establish some results concerning the topology of the represented manifolds: suspensions, fundamental groups, connected sums and moves between graphs representing the same manifold. Classification results of compact orientable 4-manifolds representable by graphs up to six vertices are obtained, together with some properties of the G-degree of 5-colored graphs relating this approach to tensor models theory.

scholarly journals Asymptotic Expansion of the Multi-Orientable Random Tensor Model

10.37236/4629 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Eric Fusy ◽  
Adrian Tanasa
Keyword(s):  
Asymptotic Expansion ◽  
Matrix Models ◽  
Three Dimensional ◽  
Natural Generalization ◽  
General Term ◽  
Tensor Model ◽  
Half Integer ◽  
Tensor Models ◽  
Random Tensor

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expanion in $N$, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.

scholarly journals TENSOR MODELS AND HIERARCHY OF n-ARY ALGEBRAS

2011 ◽  
Vol 26 (19) ◽  
pp. 3249-3258 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Algebraic Structure ◽  
The Other ◽  
Invariant Form ◽  
Infinitesimal Symmetry ◽  
Tensor Models ◽  
Symmetry Transformations

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more nonlocal infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.

scholarly journals A RENORMALIZATION PROCEDURE FOR TENSOR MODELS AND SCALAR-TENSOR THEORIES OF GRAVITY

2010 ◽  
Vol 25 (23) ◽  
pp. 4475-4492 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
General Relativity ◽  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Dynamical Variable ◽  
Renormalization Procedure ◽  
Emergent Phenomena ◽  
Tensor Models ◽  
Theories Of Gravity

Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian configurations are parametrized by a scalar and a symmetric two-tensor, it is argued that, in general situations, the infrared dynamics of the tensor models should be described by scalar-tensor theories of gravity.

TWO-DIMENSIONAL QUANTUM GRAVITY, MATRIX MODELS AND STRING THEORY

1993 ◽  
Vol 08 (07) ◽  
pp. 1185-1244 ◽  
Author(s):  
KREŠIMIR DEMETERFI
Keyword(s):  
String Theory ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Lattice Models ◽  
Field Method ◽  
Two Dimensional ◽  
Large N ◽  
The One ◽  
Low Dimensional ◽  
Lattice Approach

We review some results of the recent progress in understanding two-dimensional quantum gravity and low-dimensional string theories based on the lattice approach. The possibility to solve the lattice models exactly comes from their equivalence to large N matrix models. We describe various matrix models and their continuum limits, and discuss in some detail the phase structure of Hermitian one-matrix models. For the one-dimensional matrix model we discuss its field theoretic formulation through a collective field method and summarize some perturbative results. We compare the results obtained from matrix models to the results in the continuum approach to string theory.

A First Principles Approach for Partitioning Linear Response Properties into Additive and Cooperative Contributions

2018 ◽  
Author(s):  
Daniel Lambrecht ◽  
Eric Berquist
Keyword(s):  
First Principles ◽  
Linear Response ◽  
Building Blocks ◽  
Field Method ◽  
Response Property ◽  
Response Properties ◽  
Consistent Field ◽  
Molecular Building Blocks ◽  
Self Consistent ◽  
Self Consistent Field

We present a first principles approach for decomposing molecular linear response properties into orthogonal (additive) plus non-orthogonal/cooperative contributions. This approach enables one to 1) identify the contributions of molecular building blocks like functional groups or monomer units to a given response property and 2) quantify cooperativity between these contributions. In analogy to the self consistent field method for molecular interactions, SCF(MI), we term our approach LR(MI). The theory, implementation and pilot data are described in detail in the manuscript and supporting information.

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