scholarly journals Isosystolic inequalities and the topological expansion for random surface and matrix models

1991 ◽  
Vol 139 (1) ◽  
pp. 183-200 ◽  
Author(s):  
Vincent Rivasseau
Keyword(s):  
Matrix Models ◽  
Random Surface ◽  
Topological Expansion

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 Fermionic Matrix Models

1997 ◽  
Vol 12 (12) ◽  
pp. 2135-2291 ◽  
Author(s):  
Gordon W. Semenoff ◽  
Richard J. Szabo
Keyword(s):  
String Theory ◽  
Matrix Models ◽  
Degrees Of Freedom ◽  
Integrable Hierarchies ◽  
Gauge Theories ◽  
Target Space ◽  
Vector Model ◽  
Convergence Properties ◽  
Random Surface ◽  
Large N

We review a class of matrix models whose degrees of freedom are matrices with anti-commuting elements. We discuss the properties of the adjoint fermion one-matrix, two-matrix and gauge-invariant D-dimensional matrix models at large N and compare them with their bosonic counterparts, which are the more familiar Hermitian matrix models. We derive and solve the complete sets of loop equations for the correlators of these models and use these equations to examine critical behavior. The topological large N expansions are also constructed and their relation to other aspects of modern string theory, such as integrable hierarchies, is discussed. We use these connections to discuss the applications of these matrix models to string theory and induced gauge theories. We argue that as such the fermionic matrix models may provide a novel generalization of the discretized random surface representation of quantum gravity in which the genus sum alternates and the sums over genera for correlators have better convergence properties than their Hermitian counterparts. We discuss the use of adjoint fermions instead of adjoint scalars to study induced gauge theories. We also discuss two classes of dimensionally reduced models, a fermionic vector model and a supersymmetric matrix model, and discuss their applications to the branched polymer phase of string theories in target space dimensions D > 1 and also to the meander problem.

FERMIONS IN THE TOPOLOGICAL EXPANSION OF MATRIX MODELS

1991 ◽  
Vol 06 (09) ◽  
pp. 781-787
Author(s):  
G. FERRETTI
Keyword(s):  
Free Energy ◽  
Matrix Models ◽  
Matrix Model ◽  
Hermitian Matrix ◽  
Coupling Constant ◽  
Counting Problem ◽  
Planar Limit ◽  
Quartic Interaction ◽  
Hermitian Matrix Model ◽  
Topological Expansion

The hermitian matrix model with quartic interaction is studied in presence of fermionic variables. We obtain the contribution to the free energy due to the presence of fermions. The first two terms beyond the planar limit are explicitly found for all values of the coupling constant g. These terms represent the solution of the counting problem for vacuum diagrams with one or two fermionic loops.

LOGARITHMIC SCALING VIOLATION AND BOSE CONDENSATION IN ONE-MATRIX MODELS

1991 ◽  
Vol 06 (15) ◽  
pp. 1373-1386 ◽  
Author(s):  
CHUNG-I. TAN
Keyword(s):  
Free Energy ◽  
Matrix Models ◽  
Critical Behavior ◽  
Logarithmic Singularity ◽  
Bose Condensation ◽  
Logarithmic Scaling ◽  
Topological Expansion

In this paper, we are interested in studying critical behavior of Hermitian one-matrix models where the topological expansion for the free-energy develops a logarithmic singularity, Γ∝μ2 log μ+…. We consider models where the potential is of the type U(M)= t0V0(M)−tf log Vf(M). We show that, for tf>0, the model has an underlying fermionic structure. Continuing into the bosonic region where tf<0 allows the possibility of Bose condensation, which is shown to be responsible for a logarithmic scaling violation for the free energy.

RANDOM SURFACE ANISOTROPY AND THE MAGNETIZATION OF EPITAXIALLY GROWN THIN FILMS

1988 ◽  
Vol 49 (C8) ◽  
pp. C8-1695-C8-1696 ◽  
Author(s):  
J. R. Cullen ◽  
K. B. Hathaway
Keyword(s):  
Thin Films ◽  
Surface Anisotropy ◽  
Random Surface

scholarly journals Genus expansion of matrix models and ћ expansion of KP hierarchy

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
A. Andreev ◽  
A. Popolitov ◽  
A. Sleptsov ◽  
A. Zhabin
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Special Interest ◽  
Hermitian Matrix ◽  
Geometric Meaning ◽  
Kp Hierarchy ◽  
Hurwitz Numbers ◽  
Kp Equations ◽  
Genus Expansion ◽  
Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

scholarly journals Fermi gas approach to general rank theories and quantum curves

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Naotaka Kubo
Keyword(s):  
Matrix Models ◽  
Critical Role ◽  
Three Dimensional ◽  
Fermi Gas ◽  
Fermi Gases ◽  
Partition Functions ◽  
Density Matrices ◽  
General Rank ◽  
Chern Simons ◽  
The Ideal

Abstract It is known that matrix models computing the partition functions of three-dimensional $$ \mathcal{N} $$ N = 4 superconformal Chern-Simons theories described by circular quiver diagrams can be written as the partition functions of ideal Fermi gases when all the nodes have equal ranks. We extend this approach to rank deformed theories. The resulting matrix models factorize into factors depending only on the relative ranks in addition to the Fermi gas factors. We find that this factorization plays a critical role in showing the equality of the partition functions of dual theories related by the Hanany-Witten transition. Furthermore, we show that the inverses of the density matrices of the ideal Fermi gases can be simplified and regarded as quantum curves as in the case without rank deformations. We also comment on four nodes theories using our results.

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