scholarly journals STAR-MATRIX MODELS

1995 ◽  
Vol 10 (35) ◽  
pp. 2709-2725
Author(s):  
E. VINTELER
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Matrix Models ◽  
Critical Behavior ◽  
Gaussian Model ◽  
Explicit Formulas ◽  
Special Cases ◽  
Q Matrix

The star-matrix models are difficult to solve due to the multiple powers of the Vandermonde determinants in the partition function. We apply to these models a modified Q-matrix aprpoach and we get results consistent with those obtained by other methods. As examples we study the inhomogeneous Gaussian model on Bethe tree and matrix q-Potts-like model. For the last model in the special cases q=2 and q=3, we write down explicit formulas which determine the critical behavior of the system. For q=2 we argue that the critical behavior is indeed that of the Ising model on the ϕ3 lattice.

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.

Zeros of Partition Function and Critical Exponents of the 3D Diluted Ising Model

2016 ◽  
Vol 845 ◽  
pp. 150-153
Author(s):  
Andrey N. Vakilov
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Ising Model ◽  
Partition Function ◽  
Critical Behavior ◽  
Critical Exponents ◽  
Three Dimensional ◽  
Finite Size ◽  
Finite Size Scaling ◽  
Zeros Of Partition Function

We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.

scholarly journals KNOTS, FEYNMAN DIAGRAMS AND MATRIX MODELS

1999 ◽  
Vol 02 (03) ◽  
pp. 359-380 ◽  
Author(s):  
MARTIN GROTHAUS ◽  
LUDWIG STREIT ◽  
IGOR V. VOLOVICH
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Matrix Model ◽  
Feynman Diagrams ◽  
Generalized Function ◽  
Explicit Formulas ◽  
The Matrix ◽  
Wick Theorem ◽  
Knot Diagrams ◽  
Generalized Expectation

A U (N)-invariant matrix model with d matrix variables is studied. It was shown that in the limit N → ∞ and d→0 the model describes the knot diagrams. We realize the free partition function of the matrix model as the generalized expectation of a Hida distribution ΦN,d. This enables us to give a mathematically rigorous meaning to the partition function with interaction. For the generalized function ΦN,d, we prove a Wick theorem and we derive explicit formulas for the propagators.

scholarly journals THE BRANCHING OF GRAPHS IN 2-D QUANTUM GRAVITY

1996 ◽  
Vol 11 (29) ◽  
pp. 2351-2360
Author(s):  
M.G. HARRIS
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Quantum Gravity ◽  
Gaussian Model ◽  
Branching Ratio ◽  
Polymer Phase ◽  
Branched Polymer ◽  
Single Spin ◽  
Pure Gravity

The branching ratio is calculated for three different models of 2-D gravity, using dynamical planar ϕ3 graphs. These models are pure gravity, the D=−2 Gaussian model coupled to gravity and the single spin Ising model coupled to gravity. The ratio gives a measure of how branched the graphs dominating the partition function are. Hence it can be used to estimate the location of the branched polymer phase for the multiple Ising model coupled to 2-D gravity.

CLASSIFICATION OF CRITICAL HERMITIAN MATRIX MODELS

1991 ◽  
Vol 06 (05) ◽  
pp. 439-448 ◽  
Author(s):  
SIMON DALLEY ◽  
CLIFFORD V. JOHNSON ◽  
TIM MORRIS
Keyword(s):  
Matrix Models ◽  
Critical Behavior ◽  
Hermitian Matrix ◽  
Critical Index ◽  
Critical Conditions ◽  
Scaling Functions ◽  
Critical Properties ◽  
Special Cases ◽  
The One

The critical properties of Hermitian matrix models in the one-arc phase may be simply understood and completely classified by the behavior of the eigenvalue distribution at its ends. The most general critical behavior involves two scaling functions naturally associated with each end of the distribution, and two KdV-type string equations with differing values of the critical index m. The critical conditions are shown to include previous discoveries as special cases.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

scholarly journals Multiple phases in a generalized Gross-Witten-Wadia matrix model

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Jorge G. Russo ◽  
Miguel Tierz
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Characteristic Polynomial ◽  
Matrix Model ◽  
Phase Structure ◽  
Unitary Matrix ◽  
Two Dimensional ◽  
Toeplitz Determinants ◽  
Dimensional Parameter ◽  
Planar Limit

Abstract We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

scholarly journals The bulk, surface and corner free energies of the anisotropic triangular Ising model

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190713
Author(s):  
Rodney J. Baxter
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Partition Function ◽  
Temperature Series ◽  
Isotropic Case ◽  
Series Expansions ◽  
Exact Results ◽  
Free Energies ◽  
Bulk Surface ◽  
Bulk Free Energy

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.

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