scholarly journals Product Matrix Processes for Coupled Multi-Matrix Models and Their Hard Edge Scaling Limits

2018 ◽  
Vol 19 (9) ◽  
pp. 2599-2649 ◽  
Author(s):  
Gernot Akemann ◽  
Eugene Strahov
Keyword(s):  
Matrix Models ◽  
Scaling Limits ◽  
Hard Edge ◽  
Product Matrix

SYMMETRIC SPACE TWO-MATRIX MODELS

1992 ◽  
Vol 07 (23) ◽  
pp. 5781-5796
Author(s):  
ARLEN ANDERSON
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Symmetric Space ◽  
Matrix Models ◽  
Explicit Expression ◽  
Lie Algebras ◽  
Spherical Function ◽  
Scaling Limits ◽  
Maximum Order ◽  
Rank One

The radial form of the partition function of a two-matrix model is formally given in terms of a spherical function for matrices representing any Euclidean symmetric space. An explicit expression is obtained by constructing the spherical function by the method of intertwining. The reduction of two-matrix models based on Lie algebras is an elementary application. A model based on the rank one symmetric space isomorphic to RN is less trivial and is treated in detail. This model may be interpreted as an Ising model on a random branched polymer. It has the unusual feature that the maximum order of criticality is different in the planar and double-scaling limits.

scholarly journals Double scaling limits in gauge theories and matrix models

2006 ◽  
Vol 2006 (06) ◽  
pp. 045-045 ◽  
Author(s):  
Gaetano Bertoldi ◽  
Timothy J Hollowood ◽  
J. Luis Miramontes
Keyword(s):  
Matrix Models ◽  
Gauge Theories ◽  
Scaling Limits

scholarly journals Singular values of products of random matrices and polynomial ensembles

2014 ◽  
Vol 03 (03) ◽  
pp. 1450011 ◽  
Author(s):  
Arno B. J. Kuijlaars ◽  
Dries Stivigny
Keyword(s):  
Scaling Limit ◽  
Singular Values ◽  
Unitary Matrix ◽  
Scaling Limits ◽  
Determinantal Point Process ◽  
Double Integral ◽  
Correlation Kernel ◽  
Products Of Random Matrices ◽  
Hard Edge ◽  
The One

Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show how their result can be interpreted as a transformation of polynomial ensembles. We also show that the squared singular values of the product of M - 1 complex Ginibre matrices with one truncated unitary matrix is a polynomial ensemble, and we derive a double integral representation for the correlation kernel associated with this ensemble. We use this to calculate the scaling limit at the hard edge, which turns out to be the same scaling limit as the one found by Kuijlaars and Zhang for the squared singular values of a product of M complex Ginibre matrices. Our final result is that these limiting kernels also appear as scaling limits for the biorthogonal ensembles of Borodin with parameter θ > 0, in case θ or 1/θ is an integer. This further supports the conjecture that these kernels have a universal character.

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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